d i s s e r t a t i o n e s m a t h e m a t i c a e

P O L S K A A K A D E MI A N A U K, I N S TY TU T MA TE MA TY CZ N Y

D I S S E R T A T I O N E SM A T H E M A T I C A E(ROZPRAWY MATEMATYCZNE)

KOMITET REDAKCYJNY

ANDRZEJ BIA LYNICKI-BIRULA, BOGDAN BOJARSKI,

ZBIGNIEW CIESIELSKI, JERZY LOS,

ZBIGNIEW SEMADENI, JERZY ZABCZYK redaktor,

WIES LAW ZELAZKO zastepca redaktora

CCCLXXI

LEONID G. HANIN

Closed ideals in algebras of smooth functions

W A R S Z A W A 1998

Leonid G. HaninDepartment of MathematicsIdaho State UniversityPocatello, Idaho 83209U.S.A.E-mail: [email protected]

Published by the Institute of Mathematics, Polish Academy of Sciences

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C O N T E N T S

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. Main definitions and basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72. Closed ideals in Sobolev algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.0. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1. Preliminary observations and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2. Closed primary ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3. Spectral synthesis of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. Spectral synthesis of ideals in the algebras Cm Lipϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184. D-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215. Zygmund algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1. Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2. Extensions, approximations, and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3. Closed primary ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4. Point derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.5. An extension property and spectral synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.6. Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521. Traces of generalized Lipschitz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532. Traces of Zygmund spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583. Proof of Proposition 5.2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Acknowledgements. The author is grateful to E. M. Dyn’kin (Technion-I.I.T.) and John Wermer(Brown University) for valuable discussions.

1991 Mathematics Subject Classification: Primary 46E25, 46E35, 46H10, 46J10, 46J15, 46J20;Secondary 26A16, 41A10.

Key words and phrases: spectral synthesis of ideals, problem of spectral approximation, regularBanach algebra, point derivation, extension theorem, trace of a function, Sobolev space,Lipschitz space, Zygmund space.

Received 9.10.1996; revised version 20.2.1997.

Abstract

A topological algebra admits spectral synthesis of ideals (SSI) if every closed ideal in thisalgebra is an intersection of closed primary ideals. According to classical results this is the casefor algebras of continuous, several times continuously differentiable, and Lipschitz functions.New examples (and counterexamples) of function algebras that admit or fail to have SSI arepresented. It is shown that the Sobolev algebra W lp(Rn), 1 ≤ p <∞, has the property of SSI forand only for n = 1 and 2 ≤ n < p. It is also proved that every algebra Cm Lipϕ in one variableadmits SSI. A unified approach to SSI based on an abstract spectral synthesis theorem for aclass of Banach algebras (called D-algebras) defined in terms of point derivations and consistingof functions with “order of smoothness” not greater than 1 is discussed. Within this framework,theorems on SSI for Zygmund algebras Λϕ in one and two variables not imbedded in C1 as wellas for their separable counterparts λϕ are obtained. The fact that a Zygmund algebra Λϕ is aD-algebra is equivalent to a special extension theorem of independent interest which leads toa solution of the spectral approximation problem for the algebras Λϕ in the cases mentionedabove. Closed primary ideals and point derivations in arbitrary Zygmund algebras are described.

Introduction

A topological algebra admits spectral synthesis of ideals (SSI) if every closed proper

ideal in this algebra is an intersection of closed primary ideals. Assertions of this type

go back to classical algebraic works of E. Noether [38] and E. Lasker [29] on primary

decomposition of ideals in Noetherian rings. Later, theorems on SSI were discovered for

a number of algebras of smooth functions (for more extensive discussion, see Section 1).

As shown by G. E. Shilov [43], every proper ideal in a regular semisimple commutative

Banach algebra is an intersection of primary ideals. This shows that the collection of all

primary ideals (not necessarily closed) of a regular functional algebra is excessively ample.

Therefore, it is reasonable to confine oneself to closed primary ideals. Accordingly, every

ideal which is an intersection of closed primary ideals is a closed ideal.

Due to the influence of classical works of L. Schwartz [40], [41], A. Beurling [3], [4],

P. Malliavin [32]–[34], G. E. Shilov [43] and others, problems of spectral synthesis in

harmonic analysis attracted a lot of interest and gave rise to plenty of publications from

the late 40’s to the middle 70’s; see [2], [26], Chapter 10, [28], Chapter 5, and references

therein.

By contrast, in the nonharmonic setting, the problem of SSI is much less studied,

probably because it was commonly believed that all classical “local” algebras of smooth

functions enjoy SSI though it is usually not easy to prove. Amazingly, this is not true

even for some Sobolev algebras W lp(R

n) (see Section 2), the algebra W 22 (R2) providing

apparently the simplest known “natural” counterexample to SSI.

The circle of questions discussed in this paper is closely related to the problem of

spectral approximation, which consists in describing, for a linear topological space A of

functions defined on a topological space X, the closure JF of the set of functions in A

vanishing in a neighborhood of a given closed set F ⊂X. For any function in JF , its deri-

vatives, whatever they are, vanish in an appropriate sense on F. This renders the problem

of spectral approximation for many natural classes of functions too restrictive. Indeed, it is

not required in this problem that A is an algebra with respect to pointwise multiplication.

However, if this is the case, a more general approach developed by G. E. Shilov [43] and

based on the property of SSI is available, so that the problem of spectral approximation

reduces to characterizing minimal closed ideals with a given cospectrum.

Stimulated by Malliavin’s theorem on the lack of spectral synthesis in Wiener alge-

bras for noncompact abelian groups [32]–[34], the problem of spectral approximation was

originally studied mainly in the framework of harmonic analysis. However, profound re-

sults are also possible in the nonharmonic setting. As a most striking example, one can

mention the solution of the spectral approximation problem for Sobolev spaces [23], [24].

6 L. G. Hanin

The proofs of theorems on SSI are very individual, as a rule. An attempt at a unified

approach which enables treating a number of essentially different cases from a single

point of view is discussed in Section 4. This approach is materialized in the concept of a

D-algebra, and a theorem claiming that all D-algebras admit SSI (see [20], [22]) is stated

(see Theorem 4.1). The class of D-algebras is defined in terms of point derivations and

embraces a number of algebras of functions with the “order of smoothness” ≤ 1, among

them algebras C1 of continuously differentiable functions on a closed cube in Rn and

algebras Lip(X, ) of Lipschitz functions on a compact metric space (X, ). In these cases,

the theorem on SSI for D-algebras simplifies and unifies the proofs of known results [53],

[44], [31], [51], [17]. Furthermore, Theorem 4.1 is crucial for obtaining in Section 5 new

results on SSI for algebras Λϕ of functions on [−1, 1]n satisfying the Zygmund condition

|f(x+ h) − 2f(x) + f(x− h)| ≤ Cϕ(|h|).

The Zygmund space Λϕ is a generalization of the classical space Λ which was intro-

duced by A. Zygmund [54] and corresponds to the case n = 1, ϕ(t) = t. The space Λ and

its multivariate analogues appear in a very natural way in many problems of analysis as

the limiting space in the Lipschitz scale Lipα, 0 < α < 1, rather than Lip 1 (cf. [54]).

Despite the inclusions Lip 1 ⊂ Λ ⊂ Lipψ with ψ(t) = t log(2/t), analytic properties of

Lipschitz and Zygmund functions are significantly different, as exemplified by the almost

everywhere differentiability of functions in Lip 1 versus the existence in Λ of nowhere

differentiable functions (see e.g. [48], Chapter 5, Section 4), and also by the fact that

every real Lipschitz space Lipϕ is a lattice as opposed to the Zygmund space Λϕ which

fails to be a lattice unless it coincides with Lipϕ. As a result, Zygmund spaces are much

more difficult to handle.

The main contribution of this work consists of theorems on SSI for the Sobolev al-

gebras W lp(R

n) (Theorem 2.1), for the algebras Cm Lipϕ in one variable (Theorem 3.1),

and for the Zygmund algebras Λϕ and λϕ in one and two variables with any majorant ϕ

such that

(0)

1\0

ϕ(t)

t2dt = ∞,

or equivalently for which Λϕ is not imbedded in C1 (Theorem 5.2).

Our proof of Theorem 5.2 depends on two major assertions: on an abstract spectral

synthesis theorem (Theorem 4.1) for D-algebras [20], [22] and on a special extension

theorem (Theorem 5.1) for algebras Λϕ with n = 1, 2 and ϕ subject to condition (0). The

latter theorem claims that these algebras have an extension property (Ext) (see Section

5.4) which is equivalent to their being D-algebras, as stated in Proposition 5.5.1. We

show that whenever (Ext) holds for an algebra Λϕ with a majorant ϕ satisfying (0),

the spectral approximation problem for Λϕ has a “natural” solution (Theorem 5.3), and

that every closed ideal in the corresponding “small” Zygmund algebra λϕ is completely

determined by its cospectrum (Proposition 5.5.2).

The proof of Theorem 5.1 is based on explicit descriptions of traces of Zygmund func-

tions on an arbitrary closed subset of Rn, n = 1, 2. In the “seminormed” setting, such

descriptions were obtained in [45]. To establish the extension property (Ext), we need

Algebras of smooth functions 7

“normed” versions of these results with a control of equivalence constants; see Proposi-

tions 5.2.9 and 5.2.11. In the case n = 1, ϕ(t) = t, Theorem 5.1 was proved via a different

method in [22].

For n > 2, an intrinsic description of traces of the Zygmund spaces Λϕ is unknown.

Moreover, should such a description exist it would be insurmountably difficult [46].

Nevertheless, for the algebras Λϕ with majorant ϕ subject to (0) and satisfying certain

extra regularity conditions, the extension property (Ext) holds also for n > 2 (cf. [11]).

The proof of this result is based on the techniques of quasiharmonic extension of smooth

functions ([9], [10]) that cannot be extended to arbitrary majorants. Accordingly, under

these restrictions, this provides affirmative solution to the SSI problem for “big” and

“small” Zygmund algebras, and leads to a standard description of the sets JF in “big”

Zygmund algebras [11].

In Section 1, we give main definitions related to SSI and present basic examples of

algebras with the property of SSI.

A theorem providing complete solution to the problem of SSI for Sobolev algebras is

a matter of our concern in Section 2. This result was announced in [18].

In Section 3, we establish the property of SSI for algebras Cm Lipϕ of Cm-functions

in one variable with the derivative of order m satisfying the Lipschitz condition with

respect to a given arbitrary majorant ϕ. A natural conjecture is that this holds true

for any number of variables. However, the multivariate case is genuinely harder, and the

proof of the SSI in this case is not available to date.

The concept of D-algebras is discussed in detail in Section 4. We show that the

algebras C1 and Lip(X, ) are D-algebras, while the algebras C1 Lipϕ are not in this

class.

In Section 5, we focus our attention on Zygmund algebras. Some basic properties of

Zygmund spaces are presented in Section 5.1. Their discussion is continued in Section 5.2

with an emphasis on extension theorems, characterization of traces, and approximations.

The structure of closed primary ideals in algebras Λϕ is described in Section 5.3, whereas

Section 5.4 is devoted to description, properties, and applications of point derivations.

In a simpler particular case ϕ(t) = t, closed primary ideals and point derivations in

the Zygmund algebra Λ were characterized in [19]. Section 5.5 deals with the extension

property (Ext) for Zygmund spaces. Here, we formulate Theorem 5.1 and derive Theorems

5.2 and 5.3 as its corollaries. The proof of Theorem 5.1 is given in Section 5.6.

In the Appendix, we provide the reader with complete self-contained proofs of some

crucial facts about Zygmund algebras which are too technical to be included in our main

discussion in Section 5.

All algebras studied below are supposed to be real. However, all results can be carried

over almost word-for-word to the complex case.

1. Main definitions and basic examples

Let X be a locally compact Hausdorff space, C(X) be the space of all continuous

functions on X, and let A be a linear subspace in C(X) which is supposed to be a

8 L. G. Hanin

topological algebra with respect to pointwise multiplication of functions. We assume that

the algebra A is Shilov regular, that is, for every closed subset F of X and for each point

x ∈ X\F, there exists a function f ∈ A such that f vanishes on F and f(x) = 1. Also, it

is supposed that the space of maximal ideals of A coincides with X , i.e., every maximal

ideal in A is of the form Mx = f ∈ A : f(x) = 0 for some point x ∈ X . If X is compact

then A is assumed to contain the unity function.

An ideal I in A is a linear subspace of A such that fg ∈ I whenever f ∈ I and g ∈ A.

For every ideal I in A, we define a closed subset in X , σ(I) :=⋂f−1(0) : f ∈ I, called

the cospectrum of I. An ideal I is primary at a point x ∈ X if σ(I) = x. In other

words, a primary ideal is one contained in exactly one maximal ideal.

For each closed subset F in X , we define the set MF of all functions in A vanishing on

F , and the closure JF of the set of functions in A vanishing in a neighborhood (depending

on a function) of the set F . It follows from the above assumptions on A (see e.g. [15],

Section 36) that MF and JF are the maximal and the minimal closed ideals in A with

cospectrum F , respectively. In particular, Mx is the maximal (closed) primary ideal at x

and Jx is the minimal closed primary ideal at x. Thus, for every closed ideal I in A with

cospectrum F, one has JF ⊂ I ⊂MF .

The primary component Ix of an ideal I at a point x ∈ σ(I) is defined to be the

smallest closed primary ideal at x containing I. It is easily seen that

(1.1) Ix = closA(I + Jx).

We say that the algebra A admits SSI (notation: A ∈ Synt) if for every closed proper

ideal I in A,

(1.2) I =⋂

Ix : x ∈ σ(I).

Equivalently, A ∈ Synt if every closed proper ideal in A is an intersection of closed

primary ideals.

In the simplest case when Jx = Mx for all x ∈ X , (1.2) acquires the form

(1.3) I = MF , where F = σ(I).

This means that every closed ideal in A is completely determined by its cospectrum, and

in this case we write A ∈ synt.

In fact, the ideal MF has property (1.2) for every nonempty closed subset F of X.

The following algebras of smooth functions were known to admit SSI.

1. The algebra C(X) of all continuous functions f on a compact Hausdorff space

X supplied with the norm ‖f‖X := sup|f(x)| : x ∈ X. The fact that C(X) ∈ synt

was first established by M. Stone [49]. A simpler proof along the lines of the theory of

Banach algebras was suggested by G. E. Shilov [43], while for a proof based on a duality

argument the reader is referred e.g. to [13], Section 4.10.6. In like manner, (1.3) holds

for the algebra C0(X) of continuous functions on a locally compact Hausdorff space X

vanishing at “infinity” (see [25], Appendix C, Theorem 30).

2. The “small” Lipschitz algebra lip(X, ) consisting of functions f on a compact

metric space (X, ) with the finite norm


‖f‖X, := max‖f‖X, |f |X,,

where

|f |X, := sup

|f(x) − f(y)|

(x, y): x, y ∈ X, x 6= y

,

and satisfying the condition

(1.4) lim(x,y)→0

f(x) − f(y)

(x, y)= 0.

As shown in [42], lip(X, ) ∈ synt.

3. The “big” Lipschitz algebra Lip(X, ) of all functions on X with the finite norm

‖ · ‖X,. Note that lip(X, ) is a closed separable subalgebra in Lip(X, ). The spectral

synthesis theorem for the algebras Lip(X, ) is due to L. Waelbroeck [51]. In the particular

case of a compact subset of Rn with the metric (x, y) = |x − y|α, 0 < α ≤ 1, this fact

was established by G. Glaeser [17]. The proofs of these results are appreciably harder

as compared to the case of “small” Lipschitz algebras, mainly due to the fact that in

“big” Lipschitz algebras for every cluster point x ∈ X the quotient space Mx/Jx is

infinite-dimensional (and even nonseparable), while for “small” Lipschitz algebras this

space is trivial.

4. The algebra Cm(Ω) of m times continuously differentiable functions on an open

subsetΩ of Rn with the topology of uniform convergence of functions and their derivatives

up to order m on compact subsets of Ω. The classical theorem of H. Whitney [53] states

that Cm(Ω) ∈ Synt. In fact, the same is also true for Banach algebras Cm(Q) of Cm-

functions on a closed cube Q in Rn (for m = 1 and arbitrary n this was established in

[44], and for n = 1 with arbitrary m—in [43]) as well as for Banach algebras Cm0 (Rn) of

Cm-functions on Rn vanishing at infinity together with their derivatives of order ≤ m. A

simpler proof of Whitney’s theorem and its extension to the case m = ∞ can be found

in [31]. A similar argument yields the theorem on SSI for the algebras Cm lipϕ(Ω) of

functions in Cm(Ω) with the derivatives of order m in lipϕ(K) for every compact subset

K of Ω (cf. [18]). Here ϕ is a nondecreasing function on R+ such that ϕ(0) = ϕ(0+) = 0,

ϕ(t) > 0 for t > 0, and limt→0+ ϕ(t)/t = ∞.

Let Amn be the algebra of polynomials of degree at most m with multiplication defined

as the truncation of the usual product to degree m. For x ∈ Ω, the mapping πx : f →

Tmx f , where Tm

x f is the Taylor polynomial of order m for the function f at the point x,

identifies the quotient algebra Cm(Ω)/Jx with Amn . Hence, for every ideal I in Cm(Ω),

π−1x (πx(I)) = I + Jx is the smallest closed primary ideal in Cm(Ω) at the point x

containing I, that is,

(1.5) Ix = I + Jx

(compare with (1.1)). Also, the minimal closed primary ideal of the algebra A = Cm(Ω)

at a point x ∈ Ω has the following representation [53], [31]:

(1.6) Jx = closAMm+1x = f ∈ A : Dαf(x) = 0, |α| ≤ m.

Relation (1.5) leads to an equivalent formulation of the property of SSI for the algebras

A = Cm(Ω) (see [53], [31]):

10 L. G. Hanin

Let I be a closed ideal in A and f ∈ A. Suppose that for every point x ∈ Ω there is

a function gx ∈ I such that Tmx gx = Tm

x f . Then f ∈ I.

5. The non-quasi-analytic Denjoy–Carleman classes C(Mn) of functions in one variable

with regular weights Mn∞n=0; see [7] for details.

Examples 1–5 reveal a remarkable universality of the property of SSI in a wide range

of smoothness and topological assumptions on function algebras.

2. Closed ideals in Sobolev algebras

2.0. Notation. We denote by |x| the Euclidean norm of a vector x = (x1, . . . , xn) ∈

Rn and put Br := x ∈ R

n : |x| ≤ r. A cube Q in Rn is a set of the form Q(c, d) := x ∈

Rn : |xi − ci| ≤ d, 1 ≤ i ≤ n, and for a cube Q = Q(c, d) we write cQ := c and dQ := d.

The space Rn−1 is identified with x ∈ R

n : x1 = 0. For a multiindex α ∈ Zn+ we set

α! := α1! . . . αn! and xα := xα1

1 . . . xαnn . The symbols supp, ∇, and Dα stand, respectively,

for the support, the gradient, and the derivative of order |α| := α1 + . . . + αn (*) (the

derivatives of first order will also be denoted by Di, i = 1, . . . , n). Denote by

Tmx f(x) :=

∑

|α|≤m

Dαf(x)

α!(y − x)α

the Taylor polynomial for a function f of order m at a point x. The notation U will

be used for the closure of a subset U in Rn. Let finally mes := mesn be the Lebesgue

measure in Rn.

For a measurable subset S in Rn and for 1 ≤ p ≤ ∞ we denote by Lp(S) the space of

all measurable functions f on S with the finite norm

‖f‖p,S :=(\

S

|f(x)|p dx)1/p

(if p = ∞ we set ‖f‖∞,S := ess supx∈S |f(x)|). In the case S = Rn we shall simply write

Lp and ‖ · ‖p.

We will use the following notation for spaces of functions:

• C(S)—the space of all continuous functions on a subset S in Rn (if S = R

n we

write C(Rn) =: C);

• Cm0 —the space of all m times continuously differentiable functions on R

n vanishing

at infinity together with their derivatives up to order m;

• D—the space of C∞-functions on Rn with compact support;

•W lp, l ∈ N, 1 ≤ p <∞—the Sobolev space of functions f on R

n with the generalized

derivatives Dαf ∈ Lp, |α| ≤ l; the space W lp is supplied with the norm

‖f‖ := ‖f‖p,l :=∑

|α|≤l

‖Dαf‖p.

(*) To distinguish this norm from the Euclidean one with the same notation, we use for

multiindices Greek letters only.


Throughout Section 2, the letter A denotes positive constants (which may be different

even in the same chain of estimates) that may depend only on p, l and n.

2.1. Preliminary observations and results. The following continuous imbeddings

are classical [47], [14]:

W lp ⊂ C if p > n/l or p = n/l = 1;(2.1.1)

W lp ⊂ Lr, p ≤ r <∞ if p = n/l > 1;(2.1.2)

W lp ⊂ Ls, s

−1 = p−1 − l/n if 1 ≤ p < n/l.(2.1.3)

It follows from (2.1.1) that in the cases p > n/l and p = n/l = 1,

(2.2) ‖f‖∞ ≤ A‖f‖, f ∈ W lp.

This estimate and the density of the space D in W lp imply that in the case (2.1.1), W l

p is

continuously imbedded in C0.

Also, we note that W lp(R

n) ⊂ Lp(Rn−1) [14] in the sense that

(2.3) ‖f‖p,Rn−1 ≤ A‖f‖, f ∈ D.

It is known that in (and only in) the case (2.1.1) Sobolev spaces are algebras with respect

to pointwise multiplication. Usually, this fact is derived from the theory of multipliers for

Sobolev spaces [50], [36]. A direct proof based solely on the imbeddings (2.1.1)–(2.1.3) is

presented below (for p > n/l, a proof of the sufficiency part can be found in [1], Theorem

5.23).

Proposition 2.1.1. The Sobolev space W lp is an algebra with respect to pointwise

multiplication of functions iff p > n/l or p = n/l = 1.

Proof. Necessity. Suppose that p<n/l or p = n/l > 1. We show that the space W lp

is not an algebra.

If p < n/l, take λ such that (n/p− l)/2 ≤ λ < n/p− l. Let g be a C∞-function that

vanishes outside B1/2 and equals 1 in a neighborhood of the origin. Set f(x) := |x|−λ. It

is easy to see that Dα(|x|−λ) = Pα(x)|x|−(λ+2|α|), where Pα is a homogeneous polynomial

of order |α|. Hence, in view of (λ+ l)p < n, we have f ∈ W lp, while (2λ+ l)p ≥ n implies

f2 6∈W lp. Therefore, W l

p fails to be an algebra.

If p = n/l > 1, choose λ such that 1/(2q) ≤ λ < 1/q, and set f(x) := lnλ(1/|x|)g(x).

Note that for |α| ≥ 1,

Dα

(lnλ 1

|x|

)= |x|−2|α|

|α|∑

k=1

Pα,k(x) lnλ−k 1

|x|,

where all nonzero Pα,k are homogeneous polynomials of degree |α|, and Pα,1 6= 0. Hence,

in view of (λ − 1)p < −1, we have f ∈ W lp, while (2λ − 1)p ≥ 1 implies f2 6∈ W l

p. Thus,

in the case under study, W lp is not an algebra either.

Sufficiency. Suppose p > n/l or p = n/l = 1. To check that W lp is an algebra, it

suffices to establish the following inequality:

‖fg‖ ≤ A‖f‖ · ‖g‖, f, g ∈ D.

12 L. G. Hanin

To this end, we will show that for |α| + |β| ≤ l, |α| ≤ |β|,

(2.4) ‖Dαf ·Dβg‖p ≤ A‖f‖ · ‖g‖, f, g ∈ D.

If p > n/(l − |α|) or p = n/(l − |α|) = 1 then by (2.2),

‖Dαf ·Dβg‖p ≤ ‖Dαf‖∞‖Dβg‖p ≤ A‖Dαf‖p,l−|α|‖g‖ ≤ A‖f‖ · ‖g‖.

If 1 < p = n/(l − |α|) = n/(l − |β|) then using (2.1.2) with r = 2p we have, by the

Cauchy–Schwarz inequality,

‖Dαf ·Dβg‖p ≤ ‖Dαf‖2p‖Dβg‖2p ≤ A‖Dαf‖p,l−|α|‖g‖p,l−|β| ≤ A‖f‖ · ‖g‖.

Now consider the case 1 < p = n/(l−|α|) < n/(l−|β|). We set 1/s = 1/p− (l−|β|)/n

and r = n/(l−|β|) to obtain p/r+p/s = 1 with p < r <∞. Applying (2.1.2) and (2.1.3)

we have, using the Holder inequality,

‖Dαf ·Dβg‖p ≤ ‖Dαf‖r‖Dβg‖s ≤ A‖Dαf‖p,l−|α|‖g‖p,l−|β| ≤ A‖f‖ · ‖g‖.

In the remaining case p < n/(l − |α|) ≤ n/(l − |β|) we put 1/r1 = 1/p− (l − |α|)/n

and 1/r2 = 1/p− (l − |β|)/n and observe that p/r1 + p/r2 ≤ 2 − pl/n ≤ 1. Hence, there

are numbers t1 and t2 such that p < t1 ≤ r1, p < t2 ≤ r2 and p/t1 + p/t2 = 1. Note that

by (2.1.3),

‖Dαf‖t1 ≤ ‖Dαf‖p + ‖Dαf‖r1≤ A‖f‖,

and similarly ‖Dβg‖t2 ≤ A‖g‖. Therefore, again by the Holder inequality,

‖Dαf ·Dβg‖p ≤ ‖Dαf‖t1‖Dβg‖t2 ≤ A‖f‖ · ‖g‖,

which completes the proof of the estimate (2.4) and thereby of Proposition 2.1.1.

Remark. Let p > n/l or p = n/l = 1. It follows from Proposition 2.1.1 and from

the imbedding W lp ⊂ C0 that the Sobolev space W l

p is a Banach algebra whose space of

maximal ideals coincides with Rn.

The next assertion will be used in what follows.

Proposition 2.1.2. Let S be a measurable subset of Rn and f ∈ W l

p ∩ C. Suppose

that f |S ≡ 0, and define S0 := x ∈ S : ∇f(x) = 0. Then

mes(S \ S0) = 0.

Proof. Set Si := x ∈ S : Dif(x) = 0, i = 1, . . . , n. It suffices to show that

mes(S \ Si) = 0 for all i. Let, for example, i = 1. Write x ∈ Rn in the form x = (t, y),

where t ∈ R and y ∈ Rn−1. Denote by Sa the one-dimensional section of the set S

by the line y = a, a ∈ Rn−1. Let S′ be the (measurable) subset in R

n−1 of all points

y ∈ Rn−1 for which Sy is a nonempty measurable subset in R. Observe that for almost all

points y ∈ Rn−1 the function gy(t) := f(t, y) belongs to W l

p(R)∩C(R) and its generalized

derivative hy coincides a.e. on R with D1f |R×y. Fix any such point y. It is well known

that for almost all t ∈ R the classical derivative g′y(t) of the function gy exists and equals

hy(t). Since f(t, y) = 0 for all t ∈ Sy, we have g′y = 0 a.e. on Sy (namely, at all limit

points of Sy for which g′y exists). Hence

Algebras of smooth functions 13\S

|D1f(x)| dx =\

S′

\Sy

|hy(t)| dt dy =\

S′

\Sy

|g′y(t)| dt dy = 0,

i.e., D1f(x) = 0 for almost all x ∈ S, and Proposition 2.1.2 follows.

Proposition 2.1.3. Suppose that n = 1 ≤ p <∞ or 2 ≤ n < p <∞. Then for every

cube Q = Q(c, d) in Rn and for every function f ∈ W l

p such that f(c) = 0,

(2.5)\Q

|f(x)|p dx ≤ Adpn∑

i=1

\Q

|Dif(x)|p dx.

Proof. In the case n = p = 1 this can be checked straightforwardly. Observe that in

the remaining cases we get n/p < 1. Assume for simplicity that c = 0. We have

f(x) =

n∑

i=1

xi

1\0

Dif(tx) dt,

hence, by the Minkowski inequality for integrals,

‖f‖p,Q ≤ dn∑

i=1

1\0

‖Dif(tx)‖p,Q dt = dn∑

i=1

1\0

t−n/p‖Dif‖p,tQ dt ≤ Adn∑

i=1

‖Dif‖p,Q,

which implies (2.5).

2.2. Closed primary ideals. We assume hereafter that p > n/l or p = n/l = 1. Let

m be the maximal integer for which W lp ⊂ Cm

0 . It follows from (2.1.1) that m = l−n for

p = 1 and m is determined by the condition l− n/p− 1 ≤ m < l − n/p for p > 1.

Recall that in the algebra Cm0 the minimal closed primary ideal Jx at a point x ∈ R

n

has the form (1.6). We will show that the same is true for Sobolev algebras.

Proposition 2.2.1. Let W lp be a Sobolev algebra and m be defined as above. Then

Jx = f ∈ W lp : Dαf(x) = 0, |α| ≤ m, x ∈ R

n.

Proof. The set Hx on the right-hand side is a closed primary ideal at the point x,

hence Jx ⊂ Hx. When checking the reverse inclusion we assume without loss of generality

that x = 0. Also, it suffices to show that f ∈ H0 implies f ∈ J0 for functions f ∈ D only.

For, suppose this is true, take any function f ∈ H0, and fix ε > 0. There is a function

g ∈ D with ‖g − f‖ ≤ ε. From the continuity of the imbedding W lp ⊂ Cm

0 we conclude

that

max|α|≤m

|Dαg(0)| = max|α|≤m

|Dαg(0) −Dαf(0)| ≤ A‖g − f‖ ≤ Aε.

Let ω ∈ D be a function supported in B1 which equals 1 in a neighborhood of the

origin. For the function h := g − (Tm0 g)ω we have h ∈ H0 ∩ D and ‖h− g‖ ≤ Aε, hence

‖h− f‖ ≤ Aε. By our hypothesis, h ∈ J0. Therefore, f ∈ J0 via arbitrariness of ε.

We start with the case p 6= n/(l −m − 1). Then δ := m + n/p + 1 − l > 0. We will

show that for any function f ∈ H0 ∩ D and for all α, β ∈ Zn+ with |α| + |β| ≤ l,

(2.6) ε−|β|( \

|x|≤ε

|Dαf(x)|p dx)1/p

→ 0 as ε→ 0.

14 L. G. Hanin

Set M := sup|Dγf(x)| : x ∈ Rn, |γ| ≤ l. If |α| > m then

ε−|β|( \

|x|≤ε

|Dαf(x)|p dx)1/p

≤ AMε−|β|+n/p = AMε−|β|+δ+l−m−1

≤ AMεδ+l−|α|−|β| ≤ AMεδ.

Let now |α| ≤ m. In this case we note that Dα+γf(0) = 0 for |γ| ≤ m− |α| and use the

identity

Dαf(x) = (m− |α| + 1)∑

|γ|=m−|α|+1

xγ

γ!

1\0

Dα+γf(tx)(1 − t)m−|α| dt

to obtain

ε−|β|( \

|x|≤ε

|Dαf(x)|p dx)1/p

≤ AMε−|β|+m−|α|+1+n/p = AMεδ+l−|α|−|β| ≤ AMεδ.

Thus in both cases (2.6) holds.

For 0 < ε ≤ 1, set ωε(x) := ω(x/ε), where ω is the function defined above. It follows

from (2.6) that for |α| + |β| ≤ l one has ‖Dαf ·Dβωε‖p → 0 as ε→ 0, hence ‖fωε‖ → 0

as ε→ 0, and in combination with f − fωε ∈ J0 this yields f ∈ J0.

Now we turn to the more delicate case p = n/(l −m − 1). Observe that in this case

1 < p <∞ and δ = 0, so that the above estimates lead only to the weaker conclusion

ε−|β|( \

|x|≤ε

|Dαf(x)|pdx)1/p

≤ AM.

Therefore, ‖fωε‖ ≤ AM for 0 < ε ≤ 1.

Denote by T the set of pairs of multiindices (α, β) such that |α|+ |β| ≤ l, and let r be

the number of elements in T. Consider the Banach space Y := (Lp)r supplied with the

norm

‖F‖Y := max‖Fα,β‖p : (α, β) ∈ T , F = (Fα,β)(α,β)∈T ∈ Y.

Take a sequence εk → 0 as k → ∞, and define Fk ∈ Y by setting

Fk := (Dαf ·Dβωεk)(α,β)∈T , k ∈ N.

As was shown earlier, ‖Fk‖Y ≤ AM for all k. It follows from the reflexivity of Y that

there exist a subsequence kii∈N and F ∈ Y such that Fki→ F in the weak topology

on Y. For every k ∈ N, suppFk ⊂ Bεk, hence F = 0. Therefore, there is a sequence of

convex linear combinations

Gj =

nj∑

i=1

λijFki, λij ≥ 0,

nj∑

i=1

λij = 1,

for which Gj → 0 in the norm of Y. Now put ωj :=∑nj

i=1 λijωεki. For (α, β) ∈ T, we have

(Gj)α,β = Dαf ·Dβωj , hence ‖Dαf ·Dβωj‖p → 0 as j → ∞.

As a result, we see that in the case p = n/(l −m − 1) there is a sequence ωjj∈N

of functions in D (depending on f) such that for every j, ωj ≡ 1 in a neighborhood of

zero and ‖fωj‖ → 0 as j → ∞. Thus, f ∈ J0, and the proof of Proposition 2.2.1 is now

complete.


Remark 1. In the algebra A = W lp we have

Jx = closAMm+1x

for every x ∈ Rn (compare with (1.6)).

Remark 2. The primary component Ix of an ideal I in W lp at a point x ∈ σ(I) has

the representation (1.5).

2.3. Spectral synthesis of ideals. The main result of Section 2 is contained in the

following theorem. The proof of its sufficiency part is based on the same idea as the proof

due to B. Malgrange [31] of Whitney’s theorem on SSI for the algebras Cm(Rn) (see [53]).

Theorem 2.1. W lp ∈ Synt iff m = l − 1, i.e. iff n = 1 or 2 ≤ n < p.

Proof. Necessity. Suppose that 0 ≤ m ≤ l − 2; hence n ≥ 2, 1 ≤ p ≤ n, and l ≥ 2.

We will show that W lp 6∈ Synt.

For every compact set K in Rn, we define its p-capacity by

capp(K) := inf‖ϕ‖pp,1 : ϕ ∈ D, ϕ ≥ 1 on K.

Let E be a compact subset in Rn−1 such that mesn−1(E) > 0. If ϕ ∈ D(Rn) and

ϕ ≥ 1 on E then, in view of (2.3),

‖ϕ‖pp,1 ≥ A

\Rn−1

|ϕ(x)|p dx ≥ A\E

ϕ(x)p dx ≥ Amesn−1(E).

Therefore, capp(E) ≥ Amesn−1(E) > 0.

Take I = JE ; then I is a closed ideal in W lp, σ(I) = E, and Ix = Jx for all x ∈ E. Let

f ∈ D(Rn) be a function such that f(x) = xl−11 /(l− 1)! in a neighborhood (with respect

to Rn) of the set E. Observe that Dαf |E ≡ 0 for |α| ≤ l− 2; hence by Proposition 2.2.1,

f ∈⋂Ix : x ∈ σ(I).

We claim that f 6∈ I. In fact, should f ∈ I, then there would exist a sequence

gkk∈N ⊂ D(Rn) of functions vanishing in certain neighborhoods of E (relative to Rn) so

that gk → f in W lp as k → ∞. For functions ϕk := (∂l−1/∂xl−1

1 )(f −gk), we have ϕk → 0

in W 1p as k → ∞. From ϕk ∈ D and ϕk|E ≡ 1 we conclude that capp(E) ≤ ‖ϕk‖

pp,1,

hence capp(E) = 0. The contradiction obtained shows that f 6∈ I. Thus, property (1.1)

is missed for the ideal I, that is, W lp 6∈ Synt.

Sufficiency. Suppose that W lp ⊂ Cl−1

0 . Let I be a proper closed ideal in W lp. For

x ∈ Rn \ σ(I), set Ix := W l

p. We need to show that I =⋂Ix : x ∈ R

n. In fact, we only

have to prove that the ideal⋂Ix on the right-hand side is contained in I.

Define F := σ(I); L := dimAl−1n ; Fj := x ∈ R

n : codim Ix ≤ j, j = 0, 1, . . . , L;

E0 := F0, Ej := Fj \ Fj−1, j = 1, . . . , L. Clearly, all sets Fj are closed in Rn.

We will prove the following statement.

Lemma 2.3.1. Let f ∈⋂Ix, and let E and K be compact subsets of R

n such that

supp f ⊂ E and K ⊂ Ej ∩ E for some j. Then for every ε > 0 there exist functions

Φ ∈ D and g ∈ I so that Φ ≡ 1 in a neighborhood of K and ‖Φf − g‖ < ε.

16 L. G. Hanin

Proof. If j = L then K ∩ F = ∅. In this case take Φ to be a function in D that

vanishes in a neighborhood of F and equals 1 in a neighborhood of K, and set g := Φf.

Since Φ ∈ JF ⊂ I, we have g ∈ I. Thus, for j = L, the claim of Lemma 2.3.1 is satisfied.

Let now 0 ≤ j < L, in which case K ⊂ F. Observe that codim Ix = j for x ∈ K.

Assuming that j > 0 we find for any point x ∈ K its open neighborhood Ux and functions

v1, . . . , vj ∈ I such that T l−1y vk

jk=1 is a basis of W l

p/Jy for all y ∈ Ux ∩K. Thus,

(2.7) Dαf(y) =

j∑

k=1

ck(y)Dαvk(y), y ∈ Ux ∩K, |α| ≤ l − 1,

where c1, . . . , cj ∈ C(Ux ∩ K). In fact, the functions c1, . . . , cj may be thought of as

extended to continuous functions on K. Selecting from the covering Uxx∈K of the

compactK a finite subcovering and “glueing” the corresponding expansions (2.7) with the

help of an appropriate partition of unity, we find functions f1, . . . , fr ∈ I and λ1, . . . , λr ∈

C(K) such that

(2.8) Dαf(x) =

r∑

k=1

λk(x)Dαfk(x), x ∈ K, |α| ≤ l − 1.

Note that in the case j = 0, we have Ix = Jx for all x ∈ K, hence (2.8) is also valid with

r = 1 and λ1 ≡ 0, f1 ≡ 0.

For a ∈ K, set fa :=∑r

k=1 λk(a)fk and ha := f − fa. Obviously, fa ∈ I and

T l−1a ha = 0.

Partition Rn into equal cubes of edgelength d with disjoint interiors. Replacing every

cube by the open concentric cube of edgelength 2d we get an open covering Ui of Rn.

Let φi be a partition of unity such that φi ∈ D, suppφi ⊂ Ui, and ‖Dαφi‖∞ ≤ Ad−|α|

for all i and |α| ≤ l. For every i, define Si := k : Uk ∩ Ui 6= ∅, Vi :=⋃Uk : k ∈ Si,

and observe that Vi is the cube of edgelength 4d concentric with Ui. Set also S := i :

Ui ∩K 6= ∅, U :=⋃Ui : i ∈ S, V :=

⋃Vi : i ∈ S, and Φ :=

∑i∈S φi. For every i,

choose a point ai ∈ Ui∩K, and define g :=∑

i∈S faiφi. We readily see that Φ ∈ D, Φ ≡ 1

in a neighborhood of K, and g ∈ I.

We are going to show now that for every ε > 0 there is d > 0 such that

‖Φf − g‖ =∥∥∥

∑

i∈S

haiφi

∥∥∥ < ε.

Using the Leibniz formula we have, for |α| ≤ l,\Rn

|Dα(Φf − g)|p dx =\U

∣∣∣Dα(∑

i∈S

haiφi

)∣∣∣p

dx

≤ A∑

i∈S

\Ui

∑

k∈Si

∑

β≤α

|Dβhak|pd(|β|−|α|)p dx

≤ A∑

i∈S

∑

k∈Si

∑

β≤α

d(|β|−|α|)p\Vi

|Dβhak|p dx.

Note that for every k ∈ Si, we have ak ∈ Vi and Dγhak(ak) = 0, |γ| ≤ l − 1. Iterating

estimate (2.5) and taking into account the finiteness of multiplicity of the coverings Ui


and Vi, we get\Rn

|Dα(Φf − g)|p dx ≤ Ad(l−|α|)p∑

i∈S

∑

k∈Si

∑

|γ|=l

\Vi

|Dγhak|p dx

≤ Ad(l−|α|)p∑

i∈S

\Vi

Ψ dx ≤ Ad(l−|α|)p\V

Ψ dx,

where

Ψ :=∑

|γ|=l

(|Dγf | +M

r∑

j=1

|Dγfk|)p

with M := sup|λk(x)| : x ∈ K, 1 ≤ k ≤ r. Note that mes(V \K) → 0 as d→ 0, hence

for sufficiently small d,

‖Φf − g‖ < ε+A( \

K

Ψ dx)1/p

.

Observe that the functions f, f1, . . . , fr vanish on K. Applying Proposition 2.1.2 to suc-

cessive derivatives of these functions we conclude that Ψ(x) = 0 for almost all x ∈ K.

Therefore, ‖Φf − g‖ < ε, and Lemma 2.3.1 follows.

To complete the proof of the theorem, we have to derive from Lemma 2.3.1 the required

inclusion⋂Ix ⊂ I.

Let h ∈⋂Ix and δ > 0. Choose a function η ∈ D for which ‖h − hη‖ < δ/2. Set

f := hη and E := supp f . We show by induction that for every j = 0, 1, . . . , L the

following assertion holds:

(Pj) For every ε > 0, there exist functions Φj ∈ D and gj ∈ I such that Φj ≡ 1 in a

neighborhood of Fj ∩ E and ‖Φjf − gj‖ < ε.

Applying Lemma 2.3.1 to the set K = E0 ∩E we obtain assertion (P0). Suppose that

for some j = 1, . . . , L assertion (Pj−1) is valid. This provides us with functions Φj−1 ∈ D

and gj−1 ∈ I such that Φj−1 ≡ 1 on an open set U ⊃ Fj−1∩E and ‖Φj−1f−gj−1‖ < ε/2.

Now set K := (Fj∩E)\U. Obviously, K ⊂ Ej∩E. Applying Lemma 2.3.1 to the compact

sets E and K and to the function (1 − Φj−1)f ∈⋂Ix, we find a function Φj ∈ D which

equals 1 on an open set V ⊃ K and a function gj ∈ I such that ‖Φj(1−Φj−1)f−gj‖ < ε/2.

We set Φj := Φj−1 + Φj −Φj−1Φj and gj := gj−1 + gj to check easily that Φj ∈ D, Φj ≡ 1

in an open set U ∪ V ⊃ Fj ∩ E and gj ∈ I. It follows from the identity Φjf − gj =

[Φj(1 − Φj−1)f − gj ] + (Φj−1f − gj−1) that ‖Φjf − gj‖ < ε. Thus, assertion (Pj) holds.

Assertion (PL) with ε = δ/2 yields ‖ΦLf − gL‖ < δ, where ΦL ∈ D, ΦL ≡ 1 in a

neighborhood of E = supp f, and gL ∈ I. In view of ΦLf = f this implies ‖h− gL‖ < δ.

Recalling that the ideal I is closed we conclude via arbitrariness of δ that h ∈ I.

Theorem 2.1 is proved.

Remark 1. Theorem 2.1 implies that every Sobolev algebra with l = 1 admits SSI.

Remark 2. In the univariate case, the property of SSI in Sobolev algebras can be

established by a simpler argument using local contractions. A theorem on SSI for Sobolev

algebras of periodic functions on the line for p = 2 was stated in [39].

18 L. G. Hanin

Remark 3. For any open set Ω in Rn, Theorem 2.1 holds true for Banach algebras

(W lp)

0(Ω) defined as the closure of D(Ω) in W lp(Ω).

Remark 4. It follows from Theorem 2.1 that if n = 1 or 2 ≤ n < p then for every

closed set F in Rn,

JF = f ∈W lp(R

n) : Dαf(x) = 0, x ∈ F, |α| ≤ l − 1.

This solves the problem of spectral approximation in the case m = l− 1; see also [8]. For

arbitrary Sobolev spaces with 1 < p <∞ this problem was settled in [23], [24].

Remark 5. Suppose that m = 0. If n ≥ 2 and l ≥ 2 then by Theorem 2.1 there exist

closed subsets F in Rn such that for the Sobolev algebra W l

p(Rn) one has JF 6= IF . It

follows from the proof of Theorem 2.1 that this is the case for any compact subset F in

Rn contained in an (n − 1)-dimensional hyperplane and having there positive Lebesgue

measure.

Remark 6. For p = ∞, the Sobolev space W l∞(Rn) coincides with Cl−1 Lip 1(Rn)

and is obviously a Banach algebra whose space of maximal ideals is, however, larger than

Rn. For a similar algebra defined on a closed cube Q in R

n this difficulty does not arise.

In the next section, it will be derived from a more general result that W l∞(Q) ∈ Synt for

n = 1.

3. Spectral synthesis of ideals in the algebras Cm Lipϕ

Let ϕ : R+ → R+ be a nondecreasing function such that ϕ(0) = ϕ(0+) = 0, ϕ(t) > 0

for t > 0, and ϕ(t) = 1 for t ≥ 1. Denote by Cm Lipϕ the space of all Cm-functions f on

[0, 1] with the finite norm ‖f‖ := max‖f‖m, |f (m)|ϕ, where ‖f‖m := max‖f (k)‖[0,1] :

0 ≤ k ≤ m, and

|g|ϕ :=

sup

|g(x) − g(y)|

ϕ(|x− y|): x, y ∈ [0, 1], x 6= y

.

It can (and will) be assumed without loss of generality that the function ϕ(t)/t is nonin-

creasing for t > 0. It is easily seen that Cm Lipϕ is a Banach algebra which satisfies all

requirements of Section 1.

In Section 3, the letter A denotes various positive constants that may depend only

on m.

The following two well-known elementary statements will be used in the proof of the

spectral synthesis theorem for the algebras Cm Lipϕ.

Proposition 3.1. Let F be a compact subset of a metric space X, and let U1, . . . , Up

be a finite covering of F by open sets Ui, 1 ≤ i ≤ p. Then there exists δ > 0 such that

every subset E of X for which E ∩F 6= ∅ and diamE ≤ δ is contained in at least one of

the sets Ui, 1 ≤ i ≤ p.

Proposition 3.2. Let F be a subset of an interval [a, b] and F ′ be the set of its cluster

points. Then for every function f ∈ Cm([a, b]) such that f |F ≡ 0, we have f (k)|F ′ ≡ 0,

1 ≤ k ≤ m.


We now state the main result of Section 3.

Theorem 3.1. Cm Lipϕ ∈ Synt.

Proof. Let I be a proper closed ideal in A := Cm Lipϕ with cospectrum F. We need

to show that⋂Ix : x ∈ F ⊂ I. To this end, take f ∈

⋂Ix : x ∈ F and fix ε > 0.

In view of (1.1) for every x ∈ F there exist functions gx ∈ I, hx ∈ Jx, and rx ∈ A

such that f = gx + hx + rx, hx ≡ 0 in an open neighborhood Ux of the point x, and

‖rx‖ ≤ ε. From the covering Uxx∈F of F we extract a finite subcovering U1, . . . , Up

and take the corresponding functions gi, hi, and ri, 1 ≤ i ≤ p. By Proposition 3.1 there

is δ ∈ (0, 1] such that every set E ⊂ [0, 1] with diamE ≤ δ and E ∩ F 6= ∅ is contained

in some Ui, 1 ≤ i ≤ p.

For d > 0 consider the covering of R by the intervals

Vn := ((n− 1)d, (n+ 1)d), n ∈ Z.

Set Wn := Vn−1∪Vn∪Vn+1 = ((n−2)d, (n+2)d), n ∈ Z, and choose d so that diamWn =

4d ≤ δ. Let ψnn∈Z be a partition of unity with the properties ψn ∈ Cm+1(R),

supp ψn ⊂ [(n− 3/4)d, (n+ 3/4)d], and ‖ψ(k)n ‖∞ ≤ Ad−k, 0 ≤ k ≤ m+ 1, for all n.

Denote by F0 the set of all isolated points of F, and put

N := n ∈ Z : Vn ∩ [0, 1] 6= ∅, N0 := n ∈ N : Vn ∩ F = ∅,

N1 := n ∈ N : Vn ∩ F ′ = ∅, Vn ∩ F0 6= ∅, N2 := n ∈ N : Vn ∩ F ′ 6= ∅.

Now set fi :=∑

n∈Nifψi, i = 0, 1, 2. Then f = f0 + f1 + f2.

Obviously, f0 vanishes in a neighborhood of the set F, therefore f0 ∈ I.

Further, we claim that f1 ∈ I. To prove this, we have to show that the function f1locally belongs to I at any point x ∈ [0, 1] (i.e., coincides with a function from I in some

neighborhood of x). This is trivial for x 6∈ F. Next, f1 vanishes in a neighborhood of F ′,

hence it locally belongs to I at any point of F ′. Finally, for x ∈ F0, we use the fact that

f1 ∈ Ix to conclude (see, e.g. [43], p. 52) that f1 locally belongs to I at x.

Next we seek to prove that f2 ∈ I. If n ∈ N2 then by Proposition 3.1, Vn ⊂Wn ⊂ Uin

for some in, 1 ≤ in ≤ p. Hence

f2 =∑

n∈N2

(gin+ hin

+ rin)ψn =

∑

n∈N2

ginψn +

∑

n∈N2

rinψn = g + r,

where g :=∑

n∈N2ginψn ∈ I and r :=

∑n∈N2

rinψn. It follows from f = gin

+ hin+ rin

that rin|F ∩ Uin

≡ 0, therefore r|F ≡ 0.

We will show that ‖r‖ is small. Observe that for every n ∈ N2 there is a point

xn ∈ F ′ ∩ Vn. Then Proposition 3.2 yields r(k)(xn) = 0, 0 ≤ k ≤ m. For x ∈ Vn, we have

|r(m)(x)| = |r(m)(x) − r(m)(xn)| ≤ |r(m)|ϕϕ(|x − xn|) ≤ |r(m)|ϕϕ(2d) ≤ A|r(m)|ϕ.

Similarly, using the formula for the Taylor remainder we obtain

|r(k)(x)| ≤ Adm−k|r(m)|ϕ, 0 ≤ k ≤ m− 1.

Also, for x 6∈ V :=⋃Vn : n ∈ N2, we have r(k)(x) = 0, 0 ≤ k ≤ m. Thus,

(3.1) ‖r‖m ≤ A|r(m)|ϕ.

To estimate |r(m)|ϕ, set ∆ := |r(m)(x) − r(m)(y)|, and consider the following cases.

20 L. G. Hanin

1. If x, y 6∈ V then ∆ = 0.

2. Suppose that only one of the points x, y belongs to V, for example, x ∈ V, y 6∈ V. If

x 6∈ suppψn for all n ∈ N2 then obviously ∆ = 0. Let now x ∈ suppψn for some n ∈ N2

(note that there may exist not more than two such numbers). Since y 6∈ Vn, we have

|x− y| ≥ d/4. It follows from the Leibniz formula that

(3.2) |(rinψn)(m)(x)| ≤ A

m∑

k=0

|r(k)in

(x)|dk−m.

Applying Proposition 3.2 to the function rinvanishing on the set F∩Vn, and repeating

the argument which resulted in estimate (3.1), we readily see that for 0 ≤ k ≤ m,

(3.3) |r(k)in

(x)| ≤ Adm−kϕ(2d)|r(m)in

|ϕ ≤ Adm−kϕ(d)‖rin‖ ≤ Adm−kϕ(d)ε.

Combining this with (3.2) we obtain

∆ ≤ |r(m)(x)| = Aϕ(d)ε ≤ Aεϕ(|x− y|).

3. Suppose that the points x and y are both in V but do not belong simultaneously

to any set Vn, n ∈ N2. Then |x− y| ≥ d/2, and as in the previous case we have

∆ ≤ |r(m)(x)| + |r(m)(y)| ≤ Aεϕ(d) ≤ Aεϕ(|x − y|).

4. Suppose finally that x, y ∈ Vn0for a certain n0 ∈ N2, in which case |x − y| ≤ 2d.

Let n ∈ N2 be one of those (not more than three) numbers for which at least one of the

points x, y belongs to Vn. By the Leibniz formula

|(rinψn)(m)(x) − (rin

ψn)(m)(y)| ≤ A[ m∑

k=0

|r(k)in

(x)| · |ψ(m−k)n (x) − ψ(m−k)

n (y)|

+m∑

k=0

|r(k)in

(x) − r(k)in

(y)|dk−m].

Denote the first and the second sum by S1 and S2, respectively.

To estimate S1, observe that rin|F ∩Wn ≡ 0 and Wn ∩F ′ 6= ∅. Applying Proposition

3.2 to the function rinwe obtain (3.3). Next,

|ψ(m−k)n (x) − ψ(m−k)

n (y)| ≤ |x− y| · ‖ψ(m−k+1)n ‖∞ ≤ Adk−m+1|x− y|.

Thus in view of (3.3) and upon recalling that the function ϕ(t)/t is nonincreasing we get

S1 ≤ Aε|x− y|ϕ(d)/d ≤ Aεϕ(|x − y|).

It remains to estimate S2. For 0 ≤ k ≤ m− 1,

|r(k)in

(x) − r(k)in

(y)| = |x− y| · |rk+1in

(z)|

with some z ∈ [x, y]. Invoking (3.3) we have

|r(k)in

(x) − r(k)in

(y)| ≤ Aεdm−k−1ϕ(d)|x − y| ≤ Aεdm−kϕ(|x− y|),

which yields S2 ≤ Aεϕ(|x− y|).

Thus, in case 4 as well as in all other cases ∆ ≤ Aεϕ(|x − y|). Hence, |r(m)|ϕ ≤ Aε,

and so, due to (3.1), ‖r‖ ≤ Aε.


Recalling that I is a closed ideal we conclude from the representation f2 = g + r,

where g ∈ I, that f2 ∈ I. Therefore, f = f0 + f1 + f2 ∈ I. The proof of Theorem 3.1 is

complete.

Remark 1. The proof of Theorem 3.1 depends heavily on Proposition 3.2 and thus

cannot be carried over to the multivariate case.

Remark 2. In like manner, one can prove a theorem on SSI for the algebra

Cm Lipϕ(R), which consists of Cm-functions on R whose mth derivatives are in

Lipϕ([a, b]) for every interval [a, b], and is supplied with an appropriate family of se-

minorms.

4. D-algebras

Let A be a Shilov regular Banach algebra of continuous functions on a compact Haus-

dorff space X. We are assuming in addition that it has the following inversion property:

(4.1) If f ∈ A and f(x) 6= 0 for all x ∈ X then 1/f ∈ A.

Hence (see e.g. [43]) the space of maximal ideals of the algebra A coincides with X.

Definition 1. A bounded linear functional D ∈ A∗ is called a point derivation of the

algebra A at a point x ∈ X if

(4.2) D(fg) = f(x)Dg + g(x)Df for all f, g ∈ A.

Let Dx be the linear space of all point derivations of A at a point x ∈ X. It follows from

the above definition that Dx = (1 ∪M2x)⊥. Also, due to the regularity of A, we have

D(Jx) = 0 for all D ∈ Dx. Hence, for every isolated point x ∈ X we have Dx = 0.

For a closed subset F ⊂ X , define

(4.3) KF := D : D ∈ Dx, x ∈ F, ‖D‖ ≤ 1,

where ‖ · ‖ stands hereafter for the usual norm on A∗. Obviously, the set KF is compact

in the weak∗ topology on A∗.

With each function f ∈ A we associate a function f ∈ C(KF ) by setting

(4.4) f(D) := Df, D ∈ KF .

This formula determines a linear mapping dF : A/JF → C(KF ) which is an analogue of

the classic Gel’fand transform. We readily see that

(4.5) ‖f‖KF≤ ‖f‖F , f ∈ A,

where ‖ · ‖KFis the supremum norm on C(KF ) and ‖f‖F is the quotient norm of the

element f ∈ A/JF corresponding to f.

Now we are going to define the class of D-algebras.

Definition 2. An algebra A as above is called a D-algebra if for every closed subset

F in X there is a constant A(F ) ≥ 0 such that

(4.6) ‖f‖F ≤ A(F )‖f‖KFfor all f ∈MF .

22 L. G. Hanin

Remark 1. Equivalently, the class ofD-algebras can be characterized by the condition

that the mapping dF is an injection onto a closed subspace of C(KF ).

Remark 2. It follows from (4.6) that

(4.7) JF = closAM2F

for every closed subset F of X. In particular, Jx = closAM2x for every x ∈ X. This means

that every D-algebra consists of functions whose “order of smoothness” is less than 2.

It is intriguing that in all known examples the order of smoothness of functions in a

D-algebra is ≤ 1; see below.

Remark 3. Condition (4.6) implies that in a D-algebra for every closed subset F

of X one has JF =⋂

x∈F Jx, which means that minimal closed ideals with any given

cospectrum have the spectral synthesis property (1.2).

Remark 4. Condition (4.6) is a kind of extension theorem which implies in particular

that the “trace” of a function from a D-algebra on a closed set F ⊂ X is completely

determined by the values of the function and its point derivations at points of F .

The following result is of prime importance in the context of this paper.

Theorem 4.1. Every D-algebra admits SSI.

For the proof of this result, we refer the reader to [20], [22].

All algebras satisfying condition (1.3) fail to have nonzero point derivations and thus

provide trivial examples of D-algebras. This is the case for the algebras C(X), lip(X, ),

and W 1p (Rn) with n = 1 ≤ p <∞ and 2 ≤ n < p <∞. A number of nontrivial examples

of classical algebras which are (or fail to be) D-algebras are discussed in detail below.

(A) Algebras C1. Let C1(Q) be the algebra of C1-functions f on a closed cube Q

in Rn supplied with the norm ‖f‖ := max‖f‖Q, ‖ |∇f | ‖Q. In this algebra,

(4.8) Jx = f ∈ C1(Q) : f(x) = 0, ∇f(x) = 0, x ∈ Q,

and, in a more general way, for every closed set F ⊂ Q,

JF = f ∈ C1(Q) : f |F ≡ 0, ∇f |F ≡ 0.

It follows from (4.8) that any point derivation of the algebra C1(Q) at a point x ∈ Q has

the form f 7→ ∇f(x)ν for some vector ν ∈ Rn, i.e. is a directional derivative.

We claim that C1(Q) is a D-algebra.

To check condition (4.6), take any closed set F ⊂ Q and note that for any function

f ∈ C1(Q) vanishing on F the norm ‖f‖KFequals M := ‖ |∇f | ‖F . Given ε > 0, choose

δ > 0 such that

(4.9) |f(x)| ≤ ε and |∇f(x)| ≤M + ε, x ∈ Fδ,

where Fδ := x ∈ Q : d(x, F ) ≤ δ and d(x, F ) is the Euclidean distance from x to F.

From

(4.10) ∇f(x) −∇f(y) → 0 as x− y → 0, x, y ∈ Fδ,


and

(4.11) f(x) − f(y) −∇f(y)(x− y) = (x− y)

1\0

[∇f(y + t(x− y)) −∇f(y)] dt,

it follows that

(4.12)f(x) − f(y) −∇f(y)(x− y)

|x− y|→ 0 as x− y → 0, x, y ∈ Fδ.

In view of conditions (4.9), (4.10), and (4.12) one can apply the Whitney extension

theorem [52], [31], providing a function f ∈ C1(Q) such that f = f on Fδ and ‖f‖ ≤

A(M + ε), where A stands in Section 4 for a constant that may depend only on n. Since

f − f ∈ JF , we have ‖f‖F ≤ ‖f‖ ≤ A(M + ε), and (4.6) follows via arbitrariness of ε.

(B) Lipschitz algebras. Every real Lipschitz function defined on a subset of a metric

space (X, ) can be extended to a function from Lip(X, ) with the same Lipschitz norm;

see [37], [42]. This leads us right away to the equality

(4.13) ‖f‖F = max

‖f‖F , lim sup

x,y→F

|f(x) − f(y)|

(x, y)

, f ∈ Lip(X, )

(see [51]), where the notation x→ F means (x, F ) → 0. In particular,

JF =

f ∈ Lip(X, ) : f |F ≡ 0, lim

x,y→F

f(x) − f(y)

(x, y)= 0

;

see [42].

For a cluster point x ∈ X, denote by Φx the set of all weak* limits of linear functionals

on Lip(X, ) of the form f 7→ (f(a)−f(b))/(a, b) as a, b→ x. It is easily seen that every

functional in Φx is a point derivation at x (moreover, as shown in [42], Dx = V (Φx),

where V (·) stands for the weak* closure of the linear span). Together with (4.13) this

shows that, for real Lipschitz algebras, inequality (4.6) is satisfied with A(F ) = 1 and

hence in view of (4.5) turns into equality.

Thus, every Lipschitz algebra Lip(X, ) on a compact metric space is a D-algebra.

Point derivations play an important role in Lipschitz algebras [42]. One of their re-

markable extremal properties is stated below.

Proposition 4.1. Let D1 and D2 be point derivations of a real Lipschitz algebra

Lip(X, ) at two distinct points x1, x2 ∈ X, respectively. Then

‖D1 +D2‖ = ‖D1‖ + ‖D2‖.

Proof. We need to show that ‖D1‖+‖D2‖ ≤ ‖D1+D2‖. Fix ε > 0 and pick functions

f1, f2 ∈ Lip(X, ) such that ‖fi‖ ≤ 1 and Difi ≥ ‖Di‖ − ε, i = 1, 2. For δ ∈ (0, 1/2), set

Bi := B(xi, δ(x1, x2)), where B(x, d) is the closed ball in X of radius d and center x.

Clearly, B1 ∩ B2 = ∅. Put gi := fi − fi(xi) and define a function g on F := B1 ∪ B2 by

g|Bi = gi|Bi, i = 1, 2. An easy calculation shows that

|g|F, ≤ max

1,

2δ

1 − 2δ

.

24 L. G. Hanin

Choose δ small enough to satisfy 2δ/(1 − 2δ) ≤ 1 and δ(x1, x2) ≤ 1; then ‖g‖F, ≤ 1.

We extend functions gi, i = 1, 2, to the whole space X preserving their Lipschitz norms

(and notation) to get

‖D1‖ + ‖D2‖ ≤ D1f1 +D2f2 + 2ε = D1g1 +D2g2 + 2ε

= (D1 +D2)g + 2ε ≤ ‖D1 +D2‖ + 2ε,

and Proposition 4.1 follows.

(C) Algebras C1 Lipϕ. Let ϕ be any nondecreasing function on R+ such that ϕ(0)=

ϕ(0+) = 0 and ϕ(t)/t → ∞ as t → 0. Consider the Banach algebra A = C1 Lipϕ of

C1-functions f on a cube Q in Rn with the finite norm

‖f‖ := max‖f‖Q, ‖ |∇f | ‖Q, |∇f |ϕ,

where for a function g : Q→ Rn we set

|g|ϕ := sup

|g(x) − g(y)|

ϕ(|x− y|): x, y ∈ Q, x 6= y

.

We are going to describe closed primary ideals and point derivations of the algebra

C1 Lipϕ. As a result of this study, we shall derive that this algebra is not a D-algebra.

For x ∈ Q, denote by ‖ · ‖x the quotient norm on A/Jx, and set

Nx(f) := max

|f(x)|, |∇f(x)|, lim sup

a,b→x

|∇f(a) −∇f(b)|

ϕ(|a− b|)

, f ∈ A.

Claim 1. We have

(4.14) A‖f‖x ≤ Nx(f) ≤ ‖f‖x, f ∈ A.

In fact, for every function g ∈ A which coincides with f in a neighborhood of the

point x, we have Nx(f) = Nx(g) ≤ ‖g‖, which yields the second inequality in (4.14). To

verify the first inequality, take ε > 0 and choose δ > 0 such that

|f(y)| ≤ Nx(f) + ε, |∇f(y)| ≤ Nx(f) + ε,

and

|∇f(y) −∇f(z)| ≤ [Nx(f) + ε]ϕ(|y − z|)

whenever y, z ∈ Q and |y − x| ≤ δ, |z − x| ≤ δ. In view of (4.11) for such y and z we also

have

|f(z)− f(y) −∇f(y)(z − y)| ≤ [Nx(f) + ε]|z − y|ϕ(|y − z|).

Applying the Whitney–Glaeser extension theorem (see [16], [31] or [48], Chapter 6) to

the set F := Q(x, δ) ∩ Q we find a function f ∈ A such that f |F = f |F and ‖f‖ ≤

A[Nx(f) + ε]. Since f − f ∈ Jx, one has ‖f‖x ≤ ‖f‖ ≤ A[Nx(f) + ε]. We finish the proof

of Claim 1 by letting ε→ 0.

From (4.14) we deduce that

(4.15) Jx =

f ∈ A : f(x) = 0, ∇f(x) = 0, lim

a,b→x

∇f(a) −∇f(b)

ϕ(|a− b|)= 0

.

It follows from the condition limt→0 ϕ(t)/t = ∞ that any function in the ideal M2x

belongs to the right-hand side of (4.15). Hence, closAM2x ⊂ Jx. Recall that Jx is the


minimal closed primary ideal at x, therefore Jx ⊂ closAM2x . Thereby we have proved the

equality

(4.16) Jx = closAM2x .

Remark. For any regular commutative semisimple real Banach algebra A with unity,

(4.16) implies

(4.17) J⊥x = Dx + λδxλ∈R.

Indeed, the right-hand side of (4.17) is contained in J⊥x . To see the converse, take any

functional ψ ∈ J⊥x . Using (4.16) we have

ψ − ψ(1)δx ∈ (M2x ∪ 1)⊥ = Dx.

Therefore, ψ ∈ Dx + λδxλ∈R.

To state the next claim, define

‖f‖x := ‖f‖Kx= supDf : D ∈ Dx, ‖D‖ ≤ 1.

Claim 2. Nx(f) ≤ ‖f‖x ≤ ANx(f) for f ∈Mx.

To show this, observe that for f ∈Mx,

Nx(f) = max

|∇f(x)|, lim sup

a,b→x

|∇f(a) −∇f(b)|

ϕ(|a− b|)

.

Any functional f 7→ Deif(x), i = 1, . . . , n, is in fact a point derivation at x with norm

≤ 1. Now put

Lx(f) := lim supa,b→x

|∇f(a) −∇f(b)|

ϕ(|a− b|),

and take sequences of points ak, bk, ak 6= bk, and vectors ηk, k ∈ N, such that

|ηk| = 1 and

|Dηkf(ak) −Dηkf(bk)|

ϕ(|ak − bk|)→ Lx(f)

as k → ∞. For every k,

ψk(f) :=Dηkf(ak) −Dηkf(bk)

ϕ(|ak − bk|), f ∈ A,

is a linear continuous functional on A with norm ≤ 1. By compactness of the closed unit

ball of A∗ in the weak* topology there is a subnet kα ⊂ N and a functional ψ ∈ A∗

with ‖ψ‖ ≤ 1 such that limα ψkα(f) = ψ(f) for every f ∈ A. Using the identity

ψk(fg) = f(ak)ψk(g) + g(ak)ψk(f) +Dηkg(bk)f(bk) − f(ak)

ϕ(|ak − bk|)+Dηkf(bk)

g(bk) − g(ak)

ϕ(|ak − bk|)

and the fact that ϕ(t)/t → ∞ as t→ 0, we find that ψ satisfies (4.2) and hence is a point

derivation at x. Therefore, Nx(f) ≤ ‖f‖x.

To prove the reverse inequality, recall that D(Jx) = 0 for every D ∈ Dx. Using the

well-known dual representation of the quotient norm and Claim 1 we have

‖f‖x ≤ supλ(f) : λ ∈ J⊥x , ‖λ‖ ≤ 1 = ‖f‖x ≤ ANx(f), f ∈ A,

which completes the proof.

26 L. G. Hanin

Proposition 4.2. The algebra C1 Lipϕ is not a D-algebra.

Proof. Clearly, it suffices to show this in the case n = 1 and Q = [0, 1].

Suppose, on the contrary, that A :=C1 Lipϕ([0, 1]) is a D-algebra. Take F := 0 ∪

ak : k ∈ N, where akk∈N ⊂ [0, 1] is any positive sequence monotonically tending to

zero. Fix k and choose a function fk ∈ A which coincides with the function x 7→ x − ak

in a neighborhood of the point ak and vanishes in a neighborhood of F \ ak. Then

fk ∈MF and Nak(fk) = |f ′

k(ak)| = 1. Invoking Claim 2 one has

‖fk‖KF= sup

x∈F‖fk‖x = ‖fk‖ak

≤ A.

By the definition (4.6) of D-algebras there should exist a constant A(F ) such that for all

f ∈ A vanishing on F,

‖f‖F ≤ A(F )‖f‖KF.

Therefore,

1

ϕ(ak)=

|f ′k(ak) − f ′

k(0)|

ϕ(ak)≤ ‖fk‖F ≤ A(F ) ·A

for all k, which for k → ∞ leads to a contradiction.

(D) Zygmund algebras. The most interesting new examples of D-algebras are

found, however, among Zygmund algebras Λϕ not contained in C1, or equivalently with

ϕ satisfying condition (0). These algebras are discussed thoroughly in Section 5.

5. Zygmund algebras

5.1. Basic properties. The following notation will be used throughout Section 5.

For x ∈ Rn, we put |x| := max1≤i≤n |xi|, whereas ‖x‖ will stand for the Euclidean norm

of x. Recall that a cube in Rn is a set of the form Q(c, d) = x ∈ R

n : |x− c| ≤ d, where

c = cQ is the center of the cube and d = dQ is half of its edgelength.

For a function f defined on a convex set F in Rn and for all admissible x and h, we

put

∆1hf(x) := f(x+ h) − f(x), ∆2

hf(x) := f(x+ h) − 2f(x) + f(x− h),

and denote by

ω2(f ;F ; t) := sup|∆2hf(x)| : x± h ∈ F, |h| ≤ t, t ≥ 0,

the second modulus of continuity of f on F.

The letter A will stand for various positive constants possibly depending on n.

Let P1 be the set of all polynomials in n variables of degree not greater than 1. For

a bounded function f defined on a set F in Rn we denote by

E1(f ;F ) := inf‖P − f‖F : P ∈ P1

the best uniform approximation to f of order 1, and by P (f ;F ) any polynomial of degree

≤ 1 such that ‖P (f ;F ) − f‖F = E1(f ;F ).


It is well known [5] that for every cube Q in Rn,

(5.1) E1(f ;Q) ≤ Aω2(f ;Q; dQ)

(the reverse inequality ω2(f ;Q; dQ) ≤ 4E1(f ;Q) is obvious).

We define the Zygmund space Λϕ = Λϕ(Q0) to be the set of all bounded functions f

on the cube Q0 := [−1, 1]n satisfying, for all admissible x and h, the following Zygmund

condition:

(5.2) |∆2hf(x)| ≤ Cϕ(|h|)

with some constant C. Here, ϕ is a given nondecreasing function on R+ such that ϕ(0) =

ϕ(0+) = 0 and ϕ(t) > 0 for t > 0. Also, it will be assumed without loss of generality that

ϕ(t) = 1 for t ≥ 1 and that

(5.3)ϕ(t)

t2is nonincreasing for t > 0.

Every function ϕ with these properties will be referred to as a majorant.

Condition (5.3) implies that

ϕ(kt) ≥ k2ϕ(t), k ≥ 1,

and

(5.4) ϕ(t) ≥ t2, 0 ≤ t ≤ 1.

These inequalities will be systematically (sometimes tacitly) used in the sequel.

The space Λϕ is supplied with the norm

‖f‖Λϕ:= max‖f‖Q0

, |f |Λϕ,

where |f |Λϕ:= inf C over all C involved in (5.2).

We define the “small” Zygmund space λϕ as the set of all functions f ∈ Λϕ satisfying

the condition

(5.5) limt→0

ω2(f ;Q0; t)

ϕ(t)= 0.

A standard argument shows that λϕ is a closed separable linear subspace of Λϕ which in

the case limt→0 ϕ(t)/t2 = ∞ (in other words, for all majorants except those equivalent

to t2) can be identified with the closure of C∞(Q0) in Λϕ.

The following properties of majorants will be needed in what follows.

Proposition 5.1.1. For any majorant ϕ, the following relations hold :

s

3\s

ϕ(u)

u2du ≤ At

3\t

ϕ(u)

u2du, 0 < s ≤ t ≤ 2;(5.6.1)

s

2s\s

ϕ(u)

u2du ≤ At

2t\t

ϕ(u)

u2du, 0 < s ≤ t ≤ 1;(5.6.2)

limt→0

t

3\t

ϕ(u)

u2du = 0;(5.6.3)

28 L. G. Hanin

[t

1\t

ϕ(u)

u2du

]2

≤ Aϕ(t), 0 < t ≤ 1/2.(5.6.4)

If limt→0 ϕ(t)/t2 = ∞ then

(5.6.5) limt→0

1

ϕ(t)

[t

1\t

ϕ(u)

u2du

]2

= 0.

Proof. The first two relations depend only on the boundedness of ϕ and property

(5.3), and can be easily checked by appropriate linear changes of variables in the integrals

standing on the left-hand sides.

To show (5.6.3) take ε > 0 and choose δ > 0 such that ϕ(δ) ≤ ε. For every t ≤ δ we

have, in view of monotonicity of ϕ,

t

δ\t

ϕ(u)

u2du ≤ tϕ(δ)

(1

t−

1

δ

)≤ ϕ(δ) ≤ ε.

Therefore, for sufficiently small t,

t

3\t

ϕ(u)

u2du ≤ ε+ t

3\δ

ϕ(u)

u2du ≤ 2ε,

and (5.6.3) follows.

To prove (5.6.4) we set ω(t) := ϕ(t)/t2 and note that what we need to show is

(5.7)[ 1\

t

ω(u) du]2

≤ Aω(t), 0 < t ≤1

2.

Define

ω∗(u) :=

∞∑

k=1

ω(2−k)χ(2−k,2−k+1](u), 0 ≤ u ≤ 1,

where χE is the characteristic function of a set E. Observe that the function ω is nonin-

creasing while ω(t)t2 is nondecreasing, hence ω ≤ ω∗ ≤ 4ω. Therefore, it suffices to check

(5.4) for t = 2−n, n ∈ N, and for functions of the form

(5.8) ω =

∞∑

k=1

akχ(2−k,2−k+1] with 1 ≤ak+1

ak≤ 4.

Setting τk := ak+1/(2ak), k ∈ N, we see that under these assumptions (5.7) becomes

(5.9) (T ) :=(1 + τ1 + τ1τ2 + . . .+ τ1 · . . . · τn)2

2nτ1 · . . . · τn≤ A,

where T := (τ1, τ2, . . . , τn) and 1/2 ≤ τk ≤ 2, k = 1, . . . , n.

As a function of τ = τk, the other arguments being fixed, (T ) has the form k(τ) =

αk/τ + βk + γkτ, where αk, βk, and γk are positive constants independent of τ. The

function k(τ) is convex for τ > 0, hence its maximum on the interval [1/2, 2] is either

k(1/2) or k(2). Thus we may restrict ourselves in (5.9) to vectors T whose components

are 1/2 or 2.

Let T ∗ = (τ∗1 , . . . , τ∗n) be a vector maximizing . We claim that there exists m, 0 ≤

m ≤ n, such that τ∗i = 2 for 1 ≤ i ≤ m and τ∗i = 1/2 for m+ 1 ≤ i ≤ n. For, if not, then


for some k, 1 ≤ k ≤ n − 1, we would have τ∗k = 1/2 and τ∗k+1 = 2. Now interchanging

τ∗k and τ∗k+1 we see that an alteration on the left-hand side of (5.9) occurs only in the

(k + 1)th item τ∗1 · . . . · τ∗k−1/2 in the numerator, which becomes 2τ∗1 · . . . · τ∗k−1. Thus

for the new vector T ′ := (τ∗1 , . . . τ∗k−1, τ

∗k+1, τ

∗k , . . . , τ

∗n) we have (T ′) > (T ∗), which is a

contradiction.

It remains to note that for every m,

(T ∗) =[1 + . . .+ 2m + 2m−1 + . . .+ 2m−(n−m)]2

2n2m2−(n−m)<

(2m+1 + 2m)2

22m= 9,

and the proof of (5.6.4) is finished.

Turning to (5.6.5) observe that we only need to establish the following relation:

limn→∞

1

ω(2−n)

[ 1\2−n

ω(t) dt]2

= 0

for functions ω as in (5.8) with ak = ω(2−k). Given r ∈ N, we have by (5.9), for all n > r,

1

ω(2−n)

[ 2−r+1\2−n

ω(t) dt]2

= ω(2−r)(2−r)2(T ) = ϕ(2−r)(T ) ≤ Aϕ(2−r),

where T = (τr, . . . , τn−1). Therefore, taking into account that ϕ(t) → 0 as t → 0 we can

find (and fix) r such that

1

ω(2−n)

[ 2−r+1\2−n

ω(t) dt]2

is small enough for all n > r. Next, the condition ω(t) = ϕ(t)/t2 → ∞ as t → 0 implies

that for all sufficiently large n,

1

ω(2−n)

[ 1\2−r+1

ω(t) dt]2

will also be as small as desired.

Proposition 5.1.1 is proved.

We now state an important inequality due to Marchaud [35]: for every function f

defined on an interval L = [a− d, a+ d] and for all x, h such that x± h ∈ L and h 6= 0,

(5.10) |∆1hf(x)| ≤ A|h|

[ d\|h|

ω2(f ;L; t)

t2dt+

‖f‖L

d

].

Corollary 1. Let Q = Q(a, d) be a cube in Rn and f be a function on Q. Suppose

that F ⊂ Q and f |F ≡ 0. Then for every x ∈ Q,

(5.11) |f(x)| ≤ Ad(x, F )

[ Ad\d(x,F )

ω2(f ;Q; t)

t2dt+

‖f‖Q

d

].

Corollary 2. If f ∈ Λϕ(Q0) then for any x, y ∈ Q0, x 6= y, we have

(5.12)|f(y) − f(x)|

|y − x|≤ A‖f‖Λϕ

3\|y−x|

ϕ(t)

t2dt.

30 L. G. Hanin

In fact, by (5.10),

|f(y) − f(x)|

|y − x|≤ A

[ A\‖y−x‖

ω2(f ;Q0; t)

t2dt+ ‖f‖Q0

]≤ A‖f‖Λϕ

3\|y−x|

ϕ(t)

t2dt.

From (5.10) we conclude, invoking (5.6.3), that Λϕ ⊂ C. Moreover, as follows from

the definition (5.2) of the Zygmund space and from (5.10),

Lipϕ ⊂ Λϕ ⊂ Lipψ,

where

(5.13) ψ(t) := t

3\t

ϕ(u)

u2dt, 0 < t ≤ 1, and ψ(0) := 0.

Note also that Λϕ = Lipϕ if and only if

(5.14) t

3\t

ϕ(u)

u2du ≤ Cϕ(t), 0 < t ≤ 1,

for some constant C depending on ϕ. In particular, this is the case for ϕ(t) = tα, 0 <

α < 1.

In fact, if a majorant ϕ satisfies (5.14) then due to (5.10), Λϕ ⊂ Lipϕ, and hence

these spaces coincide.

To prove the converse, suppose that Λϕ ⊂ Lipϕ. Then this is true indeed for n = 1.

Take the odd extension to the interval [−1, 1] of the function ψ on [0, 1] defined by (5.13),

for which we preserve the same notation. As shown in the next section (see Proposition

5.2.2), ψ ∈ Λϕ([−1, 1]). Hence by our hypothesis ψ ∈ Lipϕ([−1, 1]), which implies (5.14).

Proposition 5.1.2. For every n and ϕ, the space Λϕ is a Banach algebra with respect

to pointwise multiplication.

Proof. Applying (5.10), (5.4) and (5.6.4) to the identity

(5.15) ∆2h(fg)(x) = ∆2

hf(x)g(x+ h) + 2∆1hf(x− h)∆1

hg(x) + f(x− h)∆2hg(x),

we obtain for all f, g ∈ Λϕ the following inequality:

‖fg‖Λϕ≤ A‖f‖Λϕ

‖g‖Λϕ.

Therefore, Λϕ is a Banach algebra.

Clearly, the Zygmund algebra Λϕ meets all requirements from Section 1 and has the

inversion property (4.1).

Proposition 5.1.3. The condition

(5.16)

1\0

ϕ(t)

t2dt <∞

is necessary and sufficient for the imbedding Λϕ ⊂ C1. Moreover , the latter inclusion

automatically implies Λϕ ⊂ C1 Lip γ with

γ(t) :=

t\0

ϕ(s)

s2ds.


Proof. We start with the following observation.

If f ∈ Λϕ([a, b]) then for all x, y, u, v ∈ [a, b] such that 0 < |y − x| ≤ |v − u|, we have

(5.17)

∣∣∣∣f(y) − f(x)

y − x−f(v) − f(u)

v − u

∣∣∣∣ ≤ A|f |Λϕ

2|v−u|+max|u−x|,|v−y|\|y−x|

ϕ(t)

t2dt

(for a proof of (5.17), see Proposition A2.1 in the Appendix and Remark 2 after it).

Furthermore, setting here y := x + h, u := x, v := x − h we conclude that for every

bounded function f on [a, b], (5.17) implies f ∈ Λϕ([a, b]).

Now suppose (5.16) holds. Then by (5.10) any function from Λϕ belongs to Lip 1, and

hence its partial derivatives exist almost everywhere. If in (5.17) we let y → x and v → u,

we conclude that these partial derivatives are in Lip γ and thus Λϕ ⊂ C1 Lipγ ⊂ C1.

Conversely, if Λϕ ⊂ C1 then by considering the function ψ ∈ Λϕ([−1, 1]) defined

above we obtain (5.16).


It is worth noting that for majorants ϕ satisfying the growth condition (5.16), we

have

(5.18) Λϕ = C1 Lip γ ⇔ t

t\0

ϕ(u)

u2du ≤ Cϕ(t), 0 ≤ t ≤ 1.

In particular, if limt→0 ϕ(t)/t2 < ∞ then Λϕ = C1 Lip 1, while the space λϕ is trivial,

i.e., contains only constants and linear functions.

To show (5.18), assume first that the regularity condition in (5.18) is satisfied. Take

a function f ∈ C1 Lipγ and fix x, h such that x ± h ∈ Q0 and h 6= 0. The function

g(t) := f(x+ th/|h|), t ∈ [−1, 1], obviously belongs to C1 Lip γ([−1, 1]), hence

|∆2hf(x)| =

∣∣∣|h|\0

[g′(x+ t) − g′(x− t)] dt∣∣∣

≤M

|h|\0

2t\0

ϕ(s)

s2ds dt ≤M |h|

2|h|\0

ϕ(s)

s2ds ≤ AMϕ(|h|)

with some constant M, i.e., f ∈ Λϕ. Therefore, C1 Lip γ ⊂ Λϕ, which means, owing to

Proposition 5.1.3, that Λϕ = C1 Lip γ.

Now suppose that these two spaces coincide for some n. Then, in fact, this is true for

n = 1. Clearly, γ ∈ Lip γ([−1, 1]), hence the function

f(t) :=

|t|\0

γ(s) ds =

|t|\0

ϕ(s)

s2(|t| − s) ds, |t| ≤ 1,

is in C1 Lip γ([−1, 1]) and therefore, due to our assumption, in Λϕ([−1, 1]). In particular,

f(t) = 12∆

2tf(0) ≤ Cϕ(t), 0 ≤ t ≤ 1.

Now using (5.3) we have

t

t\0

ϕ(s)

s2ds ≤ A

t/2\0

ϕ(s)

s2(t− s) ds ≤ A

t\0

ϕ(s)

s2(t− s) ds, 0 ≤ t ≤ 1,

32 L. G. Hanin

and the required regularity condition in (5.18) follows upon comparing the last three

strings of formulas.

5.2.Extensions, approximations, and traces. Collected in Section 5.2 are results

of analytic nature that will be crucial in our study of closed ideals and point derivations

in Zygmund algebras.

We start with the following simple observation (see e.g. [12]).

Proposition 5.2.1. Let f be a bounded function on an interval L = [0, d] such that

f(0) = 0, and let f be its odd extension to L := [−d, d]. Then

ω2(f ; L; t) ≤ 5ω2(f ;L; t), t ≥ 0.

Proof. When estimating ∆2hf(x) we may assume that x, x + h ∈ [0, d], h > 0, and

x− h ∈ [−d, 0). The required inequality follows from the identities

∆2hf(x) = f(x+ h) − 2f(x) − f(h− x)

= [f(h+ x) − 2f(h) + f(h− x)]

− 2[f(h− x) − 2f(h/2) + f(x)] + 2[f(h) − 2f(h/2) + f(0)]

= ∆2xf(h) − 2∆2

yf(h/2) + 2∆2h/2f(h/2)

upon observing that all points involved in the last three second differences belong to [0, d]

and that y := |x− h/2| ≤ h/2.

Corollary. Every bounded function f on an interval L = [c−d, c+d] can be extended

to a function f on L := [c− 2d, c+ 2d] in such a way that

‖f‖L ≤ A‖f‖L and ω2(f ; L) ≤ Aω2(f ;L).

Proof. Set g(t) := f(c−d+t)−f(c−d), t ∈ [0, d], and let g be the odd extension of g

to [−d, d]. Then f(x) := f(c−d)+g(x−c+d) is an extension of f to [c−2d, c+d]. Applying

the same argument to the right end of the interval [c − d, c + d] and using Proposition

5.2.1 we obtain the desired extension of the function f to the interval [c− 2d, c+ 2d].

A multivariate analogue of the corollary to Proposition 5.2.1 is contained in

Proposition 5.2.2. Let Q = Q(c, d) be a cube in Rn and f be a bounded function on

Q. There is a function f on Q := Q(c, 2d) such that f |Q = f ,

‖f‖Q ≤ A‖f‖Q, and ω2(f ; Q; t) ≤ Aω2(f ;Q; t), t ≥ 0.

This is a particular case of Proposition A1.2 in the Appendix.

In the sequel, it will be assumed when necessary that a function f ∈ Λϕ(Q) is extended

to a function f ∈ Λϕ(Q) such that ‖f‖Λϕ(Q) ≤ A‖f‖Λϕ(Q).

In the next proposition, ψ stands for the odd extension of the function defined in

(5.13) to the interval [−1, 1].

Proposition 5.2.3. ‖ψ‖Λϕ([−1,1]) ≤ A.


Proof. By Proposition 5.2.1, we have to show that ψΛϕ([0,1]) ≤ A. Note that for

0 < x ≤ 1,

x

3\x

ϕ(t)

t2dt ≤ xϕ(1)

(1

x−

1

3

)≤ 1,

therefore ‖ψ‖[0,1] ≤ 1.

When estimating ∆2hψ(x) we may assume that h > 0, which implies h < x. According

to (5.15),

∆2hψ(x) = −2h

x+h\x

ϕ(t)

t2dt+ (x− h)

[ x\x−h

ϕ(t)

t2dt−

x+h\x

ϕ(t)

t2dt

].

For the first term, I1, we have, using property (5.3) of the majorant ϕ,

|I1| = 2h

x+h\x

ϕ(t)

t2dt ≤ 2h2ϕ(x)

x2≤ 2ϕ(h).

Next, by monotonicity of ϕ,

I2 := (x− h)

[ x\x−h

ϕ(t)

t2dt−

x+h\x

ϕ(t)

t2dt

]

≤ (x− h)ϕ(x)

[(1

x− h−

1

x

)−

(1

x−

1

x+ h

)]

= 2h2 ϕ(x)

x(x + h)≤ 2h2ϕ(x)

x2≤ 2ϕ(h).

Thus, |ψ|Λϕ([−1,1]) ≤ 4, and the proof is complete.

The following extension result is of prime importance for identifying closed primary

ideals in Zygmund algebras not imbedded in C1.

Proposition 5.2.4. Suppose thatT10[ϕ(t)/t2] dt = ∞. For every δ ∈ (0, 1/2], there is

a function gδ ∈ Λϕ([−1, 1]) such that gδ(x) = x, |x| ≤ δ, and

‖gδ‖Λϕ→ 0 as δ → 0.

The proof of Proposition 5.2.4 in full generality will result from Proposition 5.2.9. It

is intriguing that for majorants ϕ satisfying the following extra regularity condition:

(5.19)

t\0

ϕ(s)

sds ≤ Aϕ(t), 0 ≤ t ≤ 1,

the required extension of the identical function can be defined by a simple explicit formula.

For ϕ(t) = t, another constructively defined extension with the same properties was found

in [19]; see also [22].

Proposition 5.2.5. Suppose that a majorant ϕ satisfies the conditions (5.19) andT10[ϕ(t)/t2] dt = ∞. Let 0 < δ ≤ 1/2 and gδ be an odd function on [−1, 1], defined by

gδ(x) := x, 0 ≤ x ≤ δ, and

34 L. G. Hanin

gδ(x) = A(δ)

[x

1\x

ϕ(t)

t2dt+

x\δ

ϕ(t)

tdt

], δ < x ≤ 1,

where A(δ) := 1/T1δ[ϕ(t)/t2] dt. Then

(5.20) ‖gδ‖Λϕ≤ A ·A(δ).

Proof. We readily see that g′δ(x) = 1 for 0 ≤ x ≤ δ and

g′δ(x) = A(δ)

1\x

[ϕ(t)/t2] dt

for δ < x ≤ 1. Hence, gδ is an increasing function, and using (5.19) with t = 1 we find

that

‖gδ‖[0,1] = gδ(1) = A(δ)

1\δ

ϕ(t)

tdt ≤ A ·A(δ).

Next, due to Proposition 5.2.1, we only need to estimate |gδ|Λϕ[0,1]. Observe that

g′′δ (x) = 0 for 0 ≤ x ≤ δ and g′′δ (x) = −A(δ)ϕ(x)/x2 for δ < x ≤ 1. For x± h ∈ [0, 1], we

have

|∆2hgδ(x)| =

∣∣∣h\0

x+t\x−t

g′′δ (s) ds dt∣∣∣ ≤ A(δ)

h\0

x+t\x−t

ϕ(s)

s2ds dt.

Since h ≤ x we obtain, in view of (5.3),

h\0

x+t\x

ϕ(s)

s2ds dt ≤

1

2h2ϕ(x)

x2≤

1

2ϕ(h).

It remains to estimate the quantity

I :=

h\0

x\x−t

ϕ(s)

s2ds dt.

We distinguish the following two cases.

Case 1: x ≥ 2h. Then

I ≤1

2h2ϕ(x − h)

(x− h)2≤

1

2ϕ(h).

Case 2: x < 2h. In this case we use condition (5.19) to find that

I =

x\x−h

[h− (x− s)]ϕ(s)

s2ds ≤

2h\0

ϕ(s)

sds ≤ Aϕ(2h) ≤ Aϕ(h).

This shows that |gδ|Λϕ≤ A ·A(δ), as required.

Remark. Let ϕ be an arbitrary majorant. For any function g on [−1, 1] extending

the identical function from the interval [−δ, δ] we have

‖g‖Λϕ≥ A

/ 1\δ

ϕ(t)

t2dt.

Therefore, the bound in (5.20) is sharp.


To check the above lower estimate, observe that if g(1) ≥ 1/2 then

‖g‖Λϕ≥ ‖g‖[−1,1] ≥ g(1) ≥

1

2≥ A

/ 1\δ

ϕ(t)

t2dt.

Now if g(1) ≤ 1/2 then setting x = u = 0, y = δ and v = 1 in (5.17), we have

‖g‖Λϕ≥ |g|Λϕ

≥ A

∣∣∣∣g(δ) − g(0)

δ − 0−g(1) − g(0)

1 − 0

∣∣∣∣/ 3\

δ

ϕ(t)

t2dt ≥ A

/ 1\δ

ϕ(t)

t2dt.

The latter remark shows that a function f from the Zygmund space Λϕ(L) on an

interval L ⊂ [−1, 1] cannot in general be extended to a function f in the Zygmund space

Λϕ on the whole interval [−1, 1] in such a way that ‖f‖Λϕ≤ C‖f‖Λϕ(L) with a constant

C independent of f and L. To have this property, one needs to renorm the space Λϕ(L).

This is done in Proposition 5.2.9 below. Moreover, the description of the trace space found

there applies to any closed subset of [−1, 1], and the property of extension uniformity

mentioned above holds in this more general case as well.

In the next proposition we assume that the function f ∈ Λϕ(Q0) is extended to

the cube 2Q0 with Zygmund norm ≤ A‖f‖Λϕ. Recall that E1(f ;Q) stands for the best

polynomial approximation of order 1 to f on a cube Q.

Proposition 5.2.6. Let f ∈ Λϕ(Q0), x ∈ Q0, and Q := Q(x, d) with 0 < d ≤ 1/2.

Suppose that for some constant M ≥ 0,

|∆2hf(y)| ≤Mϕ(|h|) for all y, h with y ± h ∈ Q(x, 2d).

Let P ∈ P1 be a polynomial satisfying ‖f − P‖Q ≤ AE1(f ;Q). Also, let ω be a C2-

function such that 0 ≤ ω ≤ 1, suppω ⊂ Q, ω ≡ 1 on Q(x, d/2), and ‖Dαω‖∞ ≤ Ad−|α|,

1 ≤ |α| ≤ 2. Then

‖(f − P )ω‖Λϕ≤ AM.

Proof. Set g := (f − P )ω and observe that, by (5.1),

‖g‖Q0≤ ‖f − P‖Q ≤ AE1(f ;Q) ≤ Aω2(f ;Q; d) ≤ AMϕ(d) ≤ AM.

When estimating ∆2hg(y) for y ± h ∈ Q0 we consider the following cases.

Case 1: |h| ≤ d/4, |y − x| ≤ d/4. Then |∆2hg(y)| = |∆2

hf(y)| ≤Mϕ(|h|).

Case 2: |h| ≤ d/4, d/4 < |y − x| ≤ 5d/4. Applying (5.15) we have

∆2hg(y) = ∆2

hf(y)ω(y + h) + 2∆1h(f − P )(y − h)∆1

hω(y) + (f − P )(y − h)∆2hω(y)

= I1 + I2 + I3.

Obviously, |I1| ≤ |∆2hf(y)| ≤Mϕ(|h|). Next, using (5.10), (5.1), and (5.3) we obtain

|I2| = 2|∆1h(f − P )(y − h)| · |∆1

hω(y)| ≤ A|h|2

d

[ Ad\‖h‖

ω2(f ;Q; d)

t2dt+

‖f − P‖Q

d

]

≤ AM|h|2

d

[ d\|h|

ϕ(t)

t2dt+

ϕ(d)

d

]≤ AM |h|2

[ϕ(|h|)

|h|2+ϕ(d)

d2

]≤ AMϕ(|h|).

36 L. G. Hanin

Finally,

|I3| ≤ A‖f − P‖Q|h|2/d2 ≤ AMϕ(d)|h|2/d2 ≤ AMϕ(|h|).

Case 3: |h| ≤ d/4, |y − x| ≥ 5d/4. In this case ∆2hg(y) = 0.

Case 4: |h| > d/4. Then |∆2hg(y)| ≤ 4‖g‖Q0

≤ AMϕ(d) ≤ AMϕ(|h|).

Summarizing we conclude that ‖(f − P )ω‖Λϕ≤ AM, as required.

Proposition 5.2.7. Suppose thatT10[ϕ(t)/t2] dt = ∞. Let F be a closed subset of Q0,

and let f be a function in Λϕ such that f |F ≡ 0. Define

M := lim supd(x,F )→0,h→0

|∆2hf(x)|

ϕ(|h|).

Then

(5.21) lim supd(x,F )→0

|f(x)|

d(x, F )T3d(x,F )

[ϕ(t)/t2] dt≤ AM.

Proof. Fix ε > 0 and choose δ > 0 such that |∆2hf(x)| ≤ (M + ε)ϕ(|h|) for all x, h

with x±h ∈ Q0, d(x, F ) ≤ δ, and |h| ≤ δ. Take a point x ∈ Q0 such that 0 < d(x, F ) ≤ δ,

and let a be a point of F with d(x, F ) = ‖x − a‖. Setting Q := Q(a, δ) and applying

inequality (5.11) we have

|f(x)| ≤ Ad(x, F )

[ δ\d(x,F )

ω2(f ;Q; t)

t2dt+

‖f‖Q

δ

]

≤ Ad(x, F )

[(M + ε)

3\d(x,F )

ϕ(t)

t2dt+

‖f‖Q0

δ

]

≤ Ad(x, F )

3\d(x,F )

ϕ(t)

t2dt

[M + ε+

‖f‖Q0

δT3d(x,F )

[ϕ(t)/t2] dt

].

Our conclusion (5.21) now follows from (0) in view of arbitrariness of ε.

The next result is an analogue of Lemma 4 in [45], which deals with the case n = 2;

compare also with Proposition 2 in [46].

Proposition 5.2.8. Let Q = Q(c, d) be a cube in Rn and f be a bounded function on

Q. Let x, y, z ∈ Q and 0 < d1 := |y − x| ≤ |z − x| := d2. Suppose that a1, . . . , an−1 are

points (in Q) such that |ai − (x+ y)/2| = d1/2, 1 ≤ i ≤ n− 1, and (ai − x)(aj − x) = 0,

1 ≤ i < j ≤ n− 1. Similarly, let points b1, . . . , bn−1 (in Q) satisfy |bi − (x+ z)/2| = d2/2,

1 ≤ i ≤ n − 1, and (bi − x)(bj − x) = 0, 1 ≤ i < j ≤ n − 1. Denote by Pxy(f) ∈ P1

the polynomial interpolating f at the points x, y, a1, . . . , an−1 and by Pxz(f) ∈ P1 the

polynomial interpolating f at the points x, z, b1, . . . , bn−1. Then

(5.22) ‖Pxy(f) − Pxz(f)‖Q ≤ Ad

2d2\d1

ω2(f ;Q; t)

t2dt.

Proof. Take r ∈ N such that 2r−1d1 ≤ d2 < 2rd1, and set Qj := Q(x, 2j−1d1),

1 ≤ j ≤ r, Qr+1 := Q(x, d2). For every j, define Pj := P (f ;Qj) − P (f ;Qj)(x) + f(x)


(recall that P (f ;S) is a polynomial of best uniform approximation of order 1 to f on a

set S). We put ω(t) := ω2(f ;Q; t) to have, by (5.1),

‖Pj − f‖Qj≤ 2‖P (f ;Qj) − f‖Qj

≤ Aω(2j−1d1), 1 ≤ j ≤ r + 1.

Next, for every g ∈ L∞(Q1),

Pxy(g)(u) =g(x) + g(y)

2+g(y) − g(x)

d21

(y − x)

(u−

x+ y

2

)(5.23)

+4

d21

n−1∑

i=1

[g(ai) −

g(x) + g(y)

2

](ai −

x+ y

2

)(u−

x+ y

2

).

Hence,

(5.24) ‖Pxy(g)‖Q1≤ A‖g‖Q1

, g ∈ L∞(Q1).

Observe now that for every cube Q(a, d) and for each polynomial P ∈ P1 such that

P (a) = 0 we have

‖P‖Q(a,λd) = λ‖P‖Q(a,d), λ ≥ 0.

Therefore, using (5.1) and (5.24) we obtain

‖Pxy(f) − P1‖Q ≤ ‖Pxy(f) − P1‖Q(x,2d) =2d

d1‖Pxy(f) − P1‖Q1

=2d

d1‖Pxy(f − P1)‖Q1

≤ Ad

d1‖f − P1‖Q1

≤ Adω(d1)

d1,

and in like manner

‖Pxz(f) − Pr+1‖Q ≤ Adω(d2)

d2.

Further, for j = 1, . . . , r we have

‖Pj − Pj+1‖Q ≤ ‖Pj − Pj+1‖Q(x,2d) ≤Ad

2jd1‖Pj − Pj+1‖Qj

≤Ad

2jd1[‖Pj − f‖Qj

+ ‖Pj+1 − f‖Qj+1]

≤Ad

2jd1[ω(2j−1d1) + ω(2jd1)] ≤ Ad

ω(2jd1)

2jd1.

Combining the above estimates we finally get

‖Pxy(f) − Pxz(f)‖Q ≤ ‖Pxy(f) − P1‖Q +

r∑

j=1

‖Pj − Pj+1‖Q + ‖Pxz(f) − Pr+1‖Q

≤ Ad

[ω(d1)

d1+

r∑

j=1

ω(2jd1)

2jd1+ω(d2)

d2

]≤ Ad

r∑

j=1

ω(2jd1)

2jd1

≤ Adr∑

j=1

2jd1\2j−1d1

ω(t)

t2dt = Ad

2rd1\d1

ω(t)

t2dt ≤ Ad

2d2\d1

ω(t)

t2dt.

The proof of Proposition 5.2.8 is finished.

Corollary. If f ∈ Λϕ then the vector field txy := ∇Pxy(f), x, y ∈ Q0, x 6= y,

satisfies the following conditions :

38 L. G. Hanin

(a) txy(y − x) = f(y) − f(x);

(b) tyx = txy;

(c) |txy| ≤ A‖f‖Λϕ

T3|y−x|

[ϕ(t)/t2] dt;

(d) if |y − x| ≤ |z − x| then

|txy − txz| ≤ A|f |Λϕ

2|z−x|\|y−x|

ϕ(t)

t2dt.

Proof. Condition (a) expresses the interpolation property of Pxy(f), equality (b)

follows from Pyx(f) = Pxy(f), relation (c) results from (5.23) and (5.12), whereas (d) is

derived from (5.22).

Let F be a closed subset in Q0. We introduce the trace space

Λϕ(F ) := f |F : f ∈ Λϕ,

and supply it with the norm

‖f‖Λϕ(F ) := inf‖f‖Λϕ: f |F = f, f ∈ Λϕ(F ),

which coincides with the quotient norm on Λϕ/MF . In contrast to the Lipschitz space

Lipϕ(F ), “intrinsic” descriptions of the Zygmund spaces Λϕ(F ) are known only for n =

1, 2 (see [45], [46]). They are formulated in Propositions 5.2.9 and 5.2.11 below, while

a more general but less constructive description valid for any n is given in Proposition

5.2.10. For proofs of Propositions 5.2.10 and 5.2.11 the reader is referred to the Appendix.

Proposition 5.2.9 (n = 1). Let F be a closed subset of [−1, 1], and f be a function

on F. Then f ∈ Λϕ(F ) iff for some M ≥ 0 the following conditions are satisfied :

(i) |f(x)| ≤M for all x ∈ F ;

(ii) for all x, y ∈ F , x 6= y,∣∣∣∣f(y) − f(x)

y − x

∣∣∣∣ ≤M

3\|y−x|

ϕ(t)

t2dt;

(iii) for all x, y, z ∈ F such that 0 < |y − x| ≤ |z − x|,

∣∣∣∣f(y) − f(x)

y − x−f(z) − f(x)

z − x

∣∣∣∣ ≤M

2|z−x|\|y−x|

ϕ(t)

t2dt.

Also,

(5.25) A1‖f‖Λϕ(F ) ≤ infM ≤ A2‖f‖Λϕ(F ), f ∈ Λϕ(F ),

with absolute constants A1 and A2.

Remark 1. For an individual function f ∈ L∞(F ), condition (iii) is necessary and

sufficient for f ∈ Λϕ(F ), and inf M in (iii) is equivalent to the seminorm |f |Λϕ(cf. [45]).

The necessity of condition (iii) is an immediate consequence of (5.22). In fact, for n = 1

the interpolation polynomial Pxy defined in Proposition 5.2.8 is simply

Pxy(f)(u) = f(x) +f(y) − f(x)

y − x(u− x),


and therefore

‖Pxy(f) − Pxz(f)‖Q = d

∣∣∣∣f(y) − f(x)

y − x−f(z) − f(x)

z − x

∣∣∣∣.

Remark 2. Condition (iii) is equivalent to (5.17) (see Proposition A2.1 in the Ap-

pendix and remarks to it).

Remark 3. For 0 < δ ≤ 1/2, the identical function on [−δ, δ] satisfies conditions

(i)–(iii) of Proposition 5.2.9 with M = maxδ, 1/T3δ[ϕ(t)/t2]dt. In view of monotonicity

of ϕ and by (5.4),3\δ

ϕ(t)

t2dt ≥ ϕ(δ)

(1

δ−

1

3

)≥Aϕ(δ)

δ≥ Aδ,

hence M ≤ A ·A(δ) (recall that A(δ) := 1/T1δ[ϕ(t)/t2] dt). The sufficiency part of Propo-

sition 5.2.9, with F = [−δ, δ], applied to f(x) = x, |x| ≤ δ, provides a function gδ ∈ Λϕ

such that ‖gδ‖Λϕ≤ A · A(δ). This justifies Proposition 5.2.4.

Proposition 5.2.10 (arbitrary n). Let F be a closed subset of [−1, 1]n, and f be a

function on F. Then f ∈ Λϕ(F ) iff for some M ≥ 0 the following conditions are satisfied :


(ii) for every pair of points x, y ∈ F , x 6= y, there exists a vector txy ∈ Rn such that

(a) txy(y − x) = f(y) − f(x);

(b) tyx = txy;

(c) |txy| ≤MT3|y−x|

[ϕ(t)/t2] dt;

(d) if 0 < |y − x| ≤ |z − x| then

|txy − txz| ≤M

2|z−x|\|y−x|

ϕ(t)

t2dt.

Also,

A1‖f‖Λϕ(F ) ≤ infM ≤ A2‖f‖Λϕ(F ), f ∈ Λϕ(F ),

where the constants A1 and A2 may depend only on n.

A “seminormed” version of Proposition 5.2.10 was stated in a slightly different form

in [46], Proposition 2.

For n = 1, Proposition 5.2.9 is derived as an immediate consequence of Proposition

5.2.10. In the case n = 2, an additional effort is needed (see Section 3 of the Appendix)

to obtain the following constructive description of the space Λϕ(F ).

Proposition 5.2.11 (n = 2). Let F be a closed subset of [−1, 1]2 not lying in a line,

and let f be a function on F. In order that f ∈ Λϕ(F ) it is necessary and sufficient that

for some M ≥ 0 the following conditions are satisfied :


(ii) for any two triples of points u = x, y, z and u′ = x′, y′, z′ in F, neither

belonging to a line, and such that 0 < |y − x| ≤ |z − x|, 0 < |y′ − x′| ≤ |z′ − x′|, and

40 L. G. Hanin

|y − x| ≤ |y′ − x′|, we have

|tu − tu′ | ≤M

[1

sinαu

2|z−x|\|y−x|

ϕ(t)

t2dt+

1

sinαu′

2|z′−x′|\|y′−x′|

ϕ(t)

t2dt

+

2|y′−x′|+max|x′−x|,|y′−y|\|y−x|

ϕ(t)

t2dt

],

where

tu :=f(y) − f(x)

‖y − x‖exy +

1

sinαu

[f(y) − f(x)

‖y − x‖−f(z) − f(x)

‖z − x‖cosαu

]e⊥xy,

exy := (y − x)/‖y − x‖, αu is the angle between the vectors exy and exz, and e⊥ stands

for the vector orthogonal to e such that the basis (e, e⊥) has standard orientation.

Also,

(5.26) A1(F )‖f‖Λϕ(F ) ≤ inf M ≤ A2‖f‖Λϕ(F ), f ∈ Λϕ(F ).

Here, A1(F ) ≤ AS(F )/d(F ), where d(F ) is the Euclidean diameter of F , S(F ) is the

area of the convex hull of F, and A2 is an absolute constant.

Remark. In fact, Proposition 5.2.11 holds true for any cube Q in R2 in place of

[−1, 1]n but the constant A1(F ) in (5.26) will depend also on ϕ(dQ). For an individual

function f ∈ L∞(F ), condition (ii) is necessary and sufficient for f ∈ Λϕ(F ) and inf M in

(ii) is equivalent to |f |Λϕwith absolute constants ([45]; see also the proof of Proposition

5.2.11 in Section 3 of the Appendix).

5.3. Closed primary ideals. We start with showing that relation (4.7) holds in the

Zygmund algebra Λϕ with any majorant ϕ (even if Λϕ is not a D-algebra) except for the

“critical” case when ϕ(t) is equivalent to t2.

Proposition 5.3.1. Suppose that limt→0 ϕ(t)/t2 = ∞. Then for every closed set

F ⊂ Q0,

JF = closΛϕM2

F .

Proof. The right-hand side of this equality is a closed ideal in Λϕ with cospectrum

F and hence contains JF . To prove the converse, it suffices to show that f2 ∈ JF for

every f ∈MF .

For δ > 0, let ωδ be a C2-function such that 0 ≤ ωδ ≤ 1, ωδ ≡ 1 on Fδ, ωδ vanishes

outside F2δ, and ‖Dαωδ‖∞ ≤ Aδ−|α|, 1 ≤ |α| ≤ 2 (such a function ωδ can be obtained

by regularization of the characteristic function of the set F3δ/2).

We claim that if f ∈ MF then ‖f2ωδ‖Λϕ→ 0 as δ → 0. In view of f2 − f2ωδ ∈ JF

this will lead us to the required inclusion closΛϕM2

F ⊂ JF .

To prove our claim, note first that ‖f2ωδ‖Q0≤ ‖f‖2

F2δ→ 0 as δ → 0. Next, for

x ± h ∈ Q0, define ∆(x, h) := |∆2h(f2ωδ)(x)|. To estimate this quantity, consider the

following cases.


Case 1: ‖h‖ ≤ δ/2, d(x, F ) ≤ δ/2. Using (5.11) and (5.7) we have

∆(x, h) = |∆2hf

2(x)| ≤ |∆2hf(x)| · [|f(x+ h)| + |f(x− h)|] + 2|∆1

hf(x− h)| · |∆1hf(x)|

≤ A|f |Λϕ‖f‖Fδ

ϕ(|h|) +A|h|2[ 1\|h|

ϕ(t)

t2dt+ ‖f‖Q0

]2

.

Case 2: ‖h‖ ≤ δ/2, δ/2 < d(x, F ) ≤ 5δ/2. In this case, by (5.15),

∆(x, h) ≤ |∆2hf

2(x)|ωδ(x + h) + 2|∆1hf

2(x− h)| · |∆1hωδ(x)|

+ |f2(x − h)| · |∆2hωδ(x)| = I1 + I2 + I3.

The first item, I1, can be estimated as in Case 1 (with F3δ in place of Fδ). Further, we

apply (5.10), (5.11), and (5.6.1) to obtain a bound for I2 as follows:

|I2| ≤ A|h|

δ|∆1

hf(x− h)| · [|f(x− h)| + |f(x)|]

≤ A|h|2[ 1\|h|

ϕ(t)

t2dt+ ‖f‖Q0

][ 1\δ

ϕ(t)

t2dt+ ‖f‖Q0

]≤ A|h|2

[ 1\|h|

ϕ(t)

t2dt+ ‖f‖Q0

]2

.

Finally, combining (5.11) and (5.6.1) we have

I3 ≤ A|h|2[ 1\|h|

ϕ(t)

t2dt+ ‖f‖Q0

]2

.

Case 3: ‖h‖ ≤ δ/2, d(x, F ) > 5δ/2. Then ∆(x, h) = 0.

Case 4: ‖h‖ > δ/2. Taking account of (5.11) and (5.6.1) we find that

∆(x, h) ≤ A‖f2‖F2δ≤ Aδ2

[ 1\2δ

ϕ(t)

t2dt+ ‖f‖Q0

]2

≤ A|h|2[ 1\|h|

ϕ(t)

t2dt+ ‖f‖Q0

]2

.

Summarizing the above estimates we see that

∆(x, h) ≤ A|f |Λϕ‖f‖F3δ

ϕ(|h|) +A|h|2[ 1\|h|

ϕ(t)

t2dt+ ‖f‖Q0

]2

.

Therefore, due to limt→0 ϕ(t)/t2 = ∞ and in view of (5.6.5),

|f2ωδ|Λϕ= sup

|∆(x, h)|

ϕ(|h|): x± h ∈ Q0

→ 0 as δ → 0.

Corollary 1. For every x ∈ Q0, Jx = closΛϕM2

x .

Corollary 2. A closed linear subspace I of Λϕ is a closed primary ideal in Λϕ if

and only if Jx ⊂ I ⊂Mx.

Proof. This is true for any commutative semisimple regular unital Banach algebra

A in which Jx = closAM2x . In fact, we have to show the “only if” part of the statement.

Suppose Jx ⊂ I ⊂Mx; we check that I is an ideal. Let f ∈ I and g ∈ Λϕ. Then

fg = f [g − g(x)] + g(x)f ∈M2x + I ⊂ Jx + I = I.

It follows from Corollary 2 that in order to describe all closed primary ideals of the

algebra Λϕ we have to characterize minimal closed primary ideals Jx. Zygmund functions

42 L. G. Hanin

can be locally approximated by linear functions. Accordingly, we start with the following

assertion.

Proposition 5.3.2. IfT10[ϕ(t)/t2] dt = ∞ then the identical function h(x) = x belongs

to the minimal closed primary ideal J0 of the algebra Λϕ([−1, 1]).

Proof. Proposition 5.2.4 provides an extension gδ of the identical function from

[−δ, δ] to [−1, 1].Note that h−gδ vanishes in the δ-neighborhood of 0 and that ‖gδ‖Λϕ→ 0

as δ → 0. Therefore, h ∈ J0.

Remark. IfT10[ϕ(t)/t2] dt < ∞ then by Proposition 5.1.3, Λϕ ⊂ C1, hence for every

function f ∈ J0 we have f ′(0) = 0. Therefore, in the case in question the identical

function does not belong to J0.

Proposition 5.3.3. (a) Suppose thatT10[ϕ(t)/t2] dt = ∞. Then the minimal closed

primary ideal of the algebra Λϕ at a point x ∈ Q0 has the form

(5.27.1) Jx =

f ∈ Λϕ : f(x) = 0, lim

y→x,h→0

∆2hf(y)

ϕ(|h|)= 0

.

(b) IfT10[ϕ(t)/t2] dt <∞ then

(5.27.2) Jx =

f ∈ Λϕ : f(x) = 0, ∇f(x) = 0, lim

y→x,h→0

∆2hf(y)

ϕ(|h|)= 0

.

Proof. The right-hand side of (5.27.1) is a closed linear subspace in Λϕ, which we

denote by Hx. The inclusion Jx ⊂ Hx is obvious. To show the converse, take f ∈ Hx and

fix ε > 0. There exists d ∈ (0, 1/2] such that

(5.28) |∆2hf(y)| ≤ εϕ(|h|) for all y, h with y ± h ∈ Q := Q(x, 2d).

Set P := P (f ;Q)− P (f ;Q)(x); then ‖f − P‖Q ≤ 2E1(f ;Q). Applying Proposition 5.2.6

we obtain for the function g := (f − P )ω the following estimate: ‖g‖Λϕ≤ Aε (for the

definion of ω, also see Proposition 5.2.6).

The polynomial P has the form P (y) =∑n

i=1 ci(yi − xi). It follows from Proposition

5.3.2 that for every i = 1, . . . , n the function y 7→ yi−xi belongs to the ideal Jx. Therefore,

P ∈ Jx. From the representation f = g+ (f − fω)+Pω, where the last two terms are in

Jx, and the norm of the first is not greater than Aε, we derive that f ∈ Jx.

To prove formula (5.27.2), it again suffices to show that its right-hand side is contained

in Jx. Take ε > 0 and choose d ∈ (0, 1] such that

|f(y)| ≤ ε and|f(y) − f(z)|

‖y − z‖≤ ε for all y, z ∈ Q := Q(x, d), y 6= z,

and (5.28) holds. Therefore, for all u, v ∈ Q, u 6= v, by (5.23) and (5.12) we have

|∇Puv(f)| ≤ Aε ≤ Aε

3\|v−u|

ϕ(t)

t2dt,

where Puv(f) is the interpolating polynomial defined in Proposition 5.2.8. Applying the

Corollary to Proposition 5.2.8 we see that for M = Aε all conditions of Proposition

5.2.10 are satisfied. Consequently, there exists a function g ∈ Λϕ such that g|Q = f |Q

and ‖g‖Λϕ≤ Aε. Since g − f ∈ Jx, we conclude that f ∈ Jx.


Remark 1. For f ∈ Λϕ, set

Nx(f) := max

|f(x)|, lim sup

y→x,h→0

|∆2hf(y)|

ϕ(|h|)

ifT10[ϕ(t)/t2] dt = ∞, and

Nx(f) := max

|f(x)|, |∇f(x)|, lim sup

y→x,h→0

|∆2hf(y)|

ϕ(|h|)

ifT10[ϕ(t)/t2] dt < ∞. If follows from Proposition 5.3.2 that Nx determines a norm on

Λϕ/Jx, and an obvious modification of its proof leads us to the conclusion that this norm

is equivalent to the quotient norm ‖ · ‖x on Λϕ/Jx:

Nx ≤ ‖ · ‖x ≤ ANx.

Remark 2. Exactly as in Proposition 4.2, we derive from (5.27.2) that in the caseT10[ϕ(t)/t2] dt <∞ the Zygmund algebra Λϕ is not a D-algebra.

5.4. Point derivations. For y ± h ∈ Q0 and h 6= 0, we set

ψy,h(f) :=∆2

hf(y)

ϕ(|h|), f ∈ Λϕ.

It is easily seen that ψy,h is a linear functional on Λϕ with norm ≤ 1. For x ∈ Q0,

denote by Ψx the set of all weak* limits of functionals ψy,h as y → x and h→ 0. It follows

from (5.15), (5.10), and (5.6.5) that in the case limt→0 ϕ(t)/t2 = ∞ every such limit is a

point derivation of Λϕ at the point x. Thus, Ψx ⊂ Dx. The structure of the space Dx of

all point derivations of the Zygmund algebra Λϕ at x ∈ Q0 is described in the following

statement.

Proposition 5.4.1. Suppose that limt→0 ϕ(t)/t2 = ∞.

(a) IfT10[ϕ(t)/t2] dt = ∞ then for every x ∈ Q0,

Dx = V (Ψx),

where V (. . .) stands for the weak* closure of the linear span.

(b) IfT10[ϕ(t)/t2] dt <∞ then

Dx = V (Ψx ∪ Dη : η ∈ Sn−1),

where Dη is the directional derivative along a vector η and Sn−1 is the unit sphere of Rn.

Proof. In case (a) we only need to check the inclusion Dx ⊂ V (Ψx). To this end,

we will show that ⊥Ψx ⊂ ⊥Dx, where ⊥L ⊂ Λϕ is the annihilator of a linear subspace L

in Λ∗ϕ.

Suppose the converse; then there is a function f ∈ ⊥Ψx \ ⊥Dx. In fact, we may

assume that f(x) = 0. Recall that in every commutative Banach algebra with unity,

Dx = (M2x ∪1)⊥. We use Corollary 1 to Proposition 5.3.1 to obtain ⊥Dx = t1t∈R +Jx,

see (4.17). Therefore, f 6∈ Jx. By Proposition 5.3.3(a) there exist ε > 0 and sequences

xm ⊂ Q0 and hm ⊂ Rn \ 0 with the properties

xm ± hm ∈ Q0, |xm − x| ≤1

m, |hm| ≤

1

m,

|∆2hmf(xm)|

ϕ(|hm|)≥ ε, m ∈ N.

44 L. G. Hanin

Put ψm := ψxm,hm, m ∈ N. Since ‖ψm‖ ≤ 1 for all m and in view of compactness

of the closed unit ball of the space Λ∗ϕ in the weak* topology, we conclude that the set

ψm : m ∈ N has a cluster point ψ. By the definition of Ψx we have ψ ∈ Ψx, hence ψ ∈ Dx.

But |ψm(f)| ≥ ε implies |ψ(f)| ≥ ε, which means that f 6∈ ⊥Ψx. The contradiction with

the choice of f shows that ⊥Ψx ⊂ ⊥Dx. This completes the proof of (a).

In case (b), the proof is essentially the same with reference to part (b) of Proposition

5.3.3.

An alternative approach to describing the subset Ψx of the set of point derivations in

Λϕ is based on making use of the Stone–Cech compactification of the topological space

Q0 := (x, h) : x ∈ Q0, h ∈ Rn \ 0, x± h ∈ Q0,

which will be denoted by K. For x ∈ Q0, let Kx be the set of all limits in K of nets

(xα, hα) ⊂ Q0 such that xα → x and hα → 0.

Proposition 5.4.2. K \ Q0 =⋃Kx : x ∈ Q0, where all sets Kx are nonempty and

disjoint.

Proof. Let ξ ∈ K \ Q0. Since Q0 is dense in K, there is a net (xα, hα) ⊂ Q0

such that (xα, hα) → ξ. Passing if necessary to a subnet we may assume that xα → x

for some x ∈ Q0. Then hα → 0. For, if not, then for some subnet β = αβ we would

have hβ → h 6= 0, and along with xβ → x this would lead us to the conclusion that

ξ = limβ(xβ , hβ) = (x, h) ∈ Q0, which contradicts the choice of ξ. Thus, ξ ∈ Kx, and

therefore, K \ Q0 =⋃Kx : x ∈ Q0.

To show that Kx is nonempty, pick for everym ∈ N points xm ∈ Q0 and hm ∈ Rn\0

so that xm±hm ∈ Q0, |xm−x| ≤ 1/m, and |hm| ≤ 1/m. It follows from the compactness

of K that there is a subnet α of N such that (xα, hα) → ξ for some ξ ∈ K. Clearly,

ξ ∈ Kx, and hence Kx 6= ∅.

Now let x and y be two distinct points of Q0. Suppose that Kx and Ky contain

a common element ξ. Then there are nets (xα, hα) and (yβ, eβ) in Q0 with the

properties (xα, hα) → ξ as xα → x, hα → 0, and (yβ , eβ) → ξ as yβ → y, eβ → 0. Set

g(z, h) :=|z − x|

|y − x|, (z, h) ∈ Q0.

The function g is continuous and bounded on Q0, hence it can be extended to a (unique)

continuous function g on K. We have g(xα, hα) → 0, therefore g(ξ) = 0. On the other

hand, g(yβ, eβ) → 1, hence g(ξ) = 1. This contradiction shows that Kx ∩Ky = ∅.


With every function f ∈ Λϕ(Q0), we associate a (uniquely defined) function f ∈ C(K)

which is the extension to K of the mapping

(5.29) (y, h) 7→∆2

hf(y)

ϕ(|h|), (y, h) ∈ Q0.

For ξ ∈ Kx, define the functional θξ(f) := f(ξ). A standard topological argument leads

us to

Proposition 5.4.3. Ψx = θξ : ξ ∈ Kx, x ∈ Q0.


We will show next that for the Zygmund algebras Λϕ with ϕ subject to condition (0),

the subspaces Dx for different x cannot be “too close” (compare with Proposition 4.1).

Proposition 5.4.4. Suppose thatT10[ϕ(t)/t2] dt = ∞. Let D1 and D2 be point de-

rivations of the algebra Λϕ at two distinct points x1 and x2 in Q0, respectively. Then

‖D1‖ + ‖D2‖ ≤ A‖D1 +D2‖.

Proof. Take δ ∈ (0, 1/2) such that the cubes Qi := Q(xi, 2δ), i = 1, 2, are disjoint.

Given ε > 0, choose functions fi ∈ Λϕ(2Q0) with ‖fi‖Λϕ≤ 1, Difi ≥ ‖Di‖ − ε, and set

Pi := P (fi;Qi), i = 1, 2. Proposition 5.2.6 with M = 1 provides us with smooth functions

ωi which equal 1 in a neighborhood of xi and vanish outside Qi and are such that for

the functions gi := (fi − Pi)ωi we have ‖gi‖Λϕ≤ A, i = 1, 2. By Proposition 5.3.2 any

linear function∑n

j=1 cj(yj −xj) belongs to Jx. Therefore, gi−fi ∈ t1t∈R +Jxi, whence

Digi = Difi, i = 1, 2. Observe also that D1g2 = D2g1 = 0. Setting g := g1 + g2 we finally

have

‖D1‖ + ‖D2‖ − 2ε ≤ D1f1 +D2f2 = D1g1 +D2g2

= (D1 +D2)g ≤ ‖D1 +D2‖ ‖g‖Λϕ≤ A‖D1 +D2‖,

and by letting ε→ 0 we obtain the required inequality.

A number of important properties of Zygmund algebras can be expressed in terms

of point derivations, as exemplified by the following three results whose counterparts for

Lipschitz algebras are presented in [42]. In the first statement, D stands for the set of all

point derivations of the algebra Λϕ.

Proposition 5.4.5. Suppose thatT10[ϕ(t)/t2] dt = ∞. Then

λϕ = ⊥D := f ∈ Λϕ : Df = 0 for all D ∈ D.

Proof. This is an immediate consequence of Proposition 5.4.1(a) and of the definition

(5.5) of the “small” Zygmund space.

Corollary. (Λϕ/λϕ)∗ is canonically isometrically isomorphic to D.

The following proposition provides a convenient criterion of weak convergence of se-

quences in the space Λϕ.

Proposition 5.4.6. Suppose that limt→0 ϕ(t)/t2 = ∞. A sequence of functions

fmm∈N ⊂ Λϕ converges to f ∈ Λϕ in the weak topology iff it satisfies the following

conditions :

(i) the set fm : m ∈ N is bounded in Λϕ;

(ii) fm(x) → f(x) as m→ ∞ for every x ∈ Q0;

(iii) Dfm → Df as m→ ∞ for every D ∈ D.

Proof. A proof is required for the sufficiency part only. Suppose conditions (i)–(iii)

are satisfied. Denote by X the disjoint union Q0 ∪K, where K is defined above as the

Stone–Cech compactification of Q0. Introduce a mapping π : Λϕ → C(X) by setting

(πf)|Q0 := f and (πf)|K := f , f ∈ Λϕ, where f is the extension to K of the function

(5.29). Indeed, π is an isometry.

46 L. G. Hanin

Take a functional η ∈ Λ∗ϕ. Then η π−1 is a bounded linear functional on the linear

subspace π(Λϕ) ⊂ C(X). By the Hahn–Banach theorem it can be extended to a bounded

linear functional on C(X). Hence there exist finite Borel measures µ ∈ M(Q0) and

ν ∈M(K) such that

η(g) =\

Q0

gdµ+\K

g dν, g ∈ Λϕ.

If (x, h) ∈ Q0 then, due to condition (ii),

fm(x, h) =∆2

hfm(x)

ϕ(|h|)→

∆2hf(x)

ϕ(|h|)= f(x, h) as m→ ∞.

Further, if ξ ∈ K \ Q0 then, by Proposition 5.4.3 and owing to (iii),

fm(ξ) = θξ(fm) → θξ(f) = f(ξ) as m→ ∞.

In view of condition (i) we can use the Lebesgue theorem on dominated convergence to

have, for m→ ∞,

η(fm) →\

Q0

f dµ+\K

f dν = η(f).

Therefore, fm → f weakly in Λϕ as m→ ∞, and Proposition 5.4.6 follows.

Our last application concerns the correspondence between the set Ix of all closed

primary ideals of the algebra Λϕ at a point x ∈ Q0 and the set Lx of all weak* closed

linear subspaces of Dx. The proof of the following statement is based on the formula⊥Dx = t1t∈R + Jx.

Proposition 5.4.7. Suppose that limt→0 ϕ(t)/t2 = ∞. The mapping

I 7→ LI := Dx ∩ I⊥, I ∈ Ix,

establishes a one-to-one correspondence between Ix and Lx. The reverse correspondence

is given by the mapping

L 7→ IL := Mx ∩⊥ L, L ∈ Lx.

Corollary. If I is an ideal in Λϕ then its primary component at a point x ∈ σ(I)

coincides with the ideal IL for L = Dx ∩ I⊥.

5.5. An extension property and spectral synthesis. We formulate the extension

property for Zygmund spaces as follows:

(Ext) For every closed subset F ⊂ Q0, for each function f ∈ Λϕ vanishing on F, and

for every ε > 0, there exist > 0 and a function g ∈ Λϕ such that g|F ≡ f |F

and ‖g‖Λϕ≤ C[NF (f) + ε], where

NF (f) := lim supd(x,F )→0, h→0

|∆2hf(x)|

ϕ(|h|),

and C is a positive constant independent of f , ε, and .

Proposition 5.5.1. Suppose thatT10[ϕ(t)/t2] dt = ∞. Then Λϕ is a D-algebra iff it

has property (Ext).


Proof. Sufficiency. Suppose Λϕ satisfies condition (Ext). Take any function f ∈ Λϕ

vanishing on a closed set F ⊂ Q0. By the definition of NF (f), there are sequences

xm ⊂ F and hm ⊂ Rn \ 0, m ∈ N, such that d(xm, F ) → 0, hm → 0, and

|∆2hmf(xm)|/ϕ(|hm|) → NF (f) as m → ∞. We may assume, in fact, that xm → x as

m→ ∞ for some point x ∈ F. For every m, the mapping f 7→ ∆2hmf(xm)/ϕ(|hm|) defines

a bounded linear functional ψm on Λϕ with ‖ψm‖ ≤ 1. Hence, there is a subnet mα

of N and a functional ψ ∈ Λ∗ϕ with ‖ψ‖ ≤ 1 such that ψm → ψ in the weak* topology

on Λ∗ϕ. It follows from our assumption (0) that limt→0 ϕ(t)/t2 = ∞. We argue as at

the beginning of Section 5.4 to find that ψ is a point derivation of Λϕ at the point x.

Therefore (see (4.3) and (4.4)),

NF (f) = limα

|ψmα(f)| = |ψ(f)| = |f(ψ)| ≤ ‖f‖KF

.

Let g be the function provided for a given ε > 0 by (Ext). Since g − f ∈ JF , we have

‖f‖F ≤ ‖g‖Λϕ≤ C[NF (f) + ε]. Hence, due to arbitrariness of ε,

‖f‖F ≤ CNF (f) ≤ C‖f‖KF,

where C may depend only on F , n, and ϕ. Thus, Λϕ is a D-algebra.

Necessity. Suppose Λϕ is a D-algebra. By Remark 1 to Proposition 5.3.3 we have, for

every x ∈ F, D ∈ Dx with ‖D‖ ≤ 1, and f ∈MF ,

|Df | ≤ ‖f‖x ≤ ANx(f) ≤ ANF (f),

hence ‖f‖KF≤ ANF (f). Combining this with the definition (4.6) of a D-algebra we

see that ‖f‖F ≤ CNF (f), where C := A · A(F ). This shows that for every ε > 0 there

exists a function k ∈ Λϕ vanishing in a -neighborhood of the set F and such that

‖f + k‖Λϕ≤ C[NF (f) + ε]. Therefore, g := f + k is the function required in (Ext).


Proposition 5.5.2. LetT10[ϕ(t)/t2] dt = ∞. Suppose that (Ext) holds. Then λϕ ∈

synt.

Proof. To establish property (1.3) for the algebra λϕ we have to show that MF ⊂ JF

for every closed subset F of Q0. Take f ∈ MF and fix ε > 0. We may think of f as

being extended to a function in λϕ(2Q0) with norm ≤ A‖f‖Λϕ(see Proposition 5.2.2 for

substantiation). Preserving the notation f for this function, we have NF (f) = 0. Hence

by (Ext) there is a number > 0 and a function g ∈ Λϕ(2Q0) such that g|F ≡ f |F and

‖g‖Λϕ(2Q0) ≤ ε. For the function k := g − f, we have k|F ≡ 0 and ‖f − k‖Λϕ(2Q0) ≤ ε.

Let w be a C∞-function with the properties w ≥ 0, suppw ⊂ Q0, andT

Rn w(x) dx = 1.

Set wδ(x) := δ−nw(x/δ), δ ∈ (0, 1]. For u ∈ Λϕ(2Q0), consider the mollified functions

uδ(x) :=\

Rn

u(x− t)wδ(t) dt.

It is easy to see that uδ ∈ C∞(Q0) ⊂ λϕ(Q0) and ‖uδ‖Λϕ≤ ‖u‖Λϕ

.

Since f ∈ λϕ(2Q0), there exists σ ∈ (0, 1] such that |∆2hf(x)| ≤ εϕ(|h|) for all x ∈ 2Q0

and |h| ≤ σ with x± h ∈ 2Q0. Therefore, for any δ ∈ (0, 1],

(5.30) |∆2h(f − fδ)(x)| ≤ 2εϕ(|h|), x ∈ Q0, |h| ≤ σ.

48 L. G. Hanin

Further, it follows from the continuity of f that there is τ ∈ (0, 1] such that for all

δ ∈ (0, τ ] we have ‖f − fδ‖Q0≤ εϕ(σ). Hence, for any |h| > σ and x ∈ Q0,

(5.31) |∆2h(f − fδ)(x)| ≤ 4‖f − fδ‖Q0

≤ 4εϕ(σ) ≤ 4εϕ(|h|), δ ∈ (0, τ ].

Combining (5.30) and (5.31) we see that ‖f − fδ‖Λϕ≤ 4ε whenever δ ∈ (0, τ ] and

therefore, for such δ,

‖f − kδ‖Λϕ≤ ‖f − fδ‖Λϕ

+ ‖fδ − kδ‖Λϕ≤ 4ε+ ‖f − k‖Λϕ

≤ 5ε.

Observe finally that if 0 < δ < min, τ then the function kδ vanishes in a neighborhood

of the set F. Therefore, f ∈ JF , which completes the proof.

We are now in a position to state the main results of Section 5.

Theorem 5.1. For n = 1, 2 and for any majorant ϕ such thatT10[ϕ(t)/t2] dt = ∞, the

algebra Λϕ possesses property (Ext).

The proof of Theorem 5.1 will be given in Section 5.6.

Our next result is a combination of Theorem 4.1 with Theorem 5.1, Proposition 5.5.1,

and Proposition 5.5.2.

Theorem 5.2. Suppose that n = 1, 2 andT10[ϕ(t)/t2] dt = ∞. Then Λϕ ∈ Synt and

λϕ ∈ synt.

As another corollary to Theorem 5.1 we obtain the following solution to the spectral

approximation problem for Zygmund algebras not imbedded in C1.

Theorem 5.3. Suppose that n = 1, 2 andT10[ϕ(t)/t2] dt = ∞. Then for any closed

subset F of Q0, the quotient norm on MF/JF is equivalent to NF :

NF (f) ≤ ‖f‖F ≤ CNF (f), f ∈MF ,

where C is the constant specified in (Ext).

In particular ,

JF =

f ∈ Λϕ : f |F ≡ 0 and lim

d(x,F )→0,h→0

∆2hf(x)

ϕ(|h|)= 0

.

It is shown in [11] that if a majorant ϕ satisfies along with (0) the following two extra

regularity conditions:t\0

ϕ(s)

sds ≤ Cϕ(t), 0 ≤ t ≤ 1,

t21\t

ϕ(s)

s3ds ≤ Cϕ(t), 0 < t ≤ 1,

then Theorems 5.1–5.3 are true also for n > 2.

5.6. Proof of Theorem 5.1. First, we are going to check (Ext) for n = 1 proceeding

from Proposition 5.2.9. Then we will combine the constructive part of our univariate

argument with Proposition 5.2.11 to obtain (Ext) in the case n = 2.


5.6.1. n = 1. Let F be a closed subset of [−1, 1] and f be a function in Λϕ vanishing

on F. By the definition of the number NF (f) and in view of Proposition 5.2.7, given ε > 0

one can find δ ∈ (0, 1] such that ‖f‖Fδ≤ ε,

(5.32) |∆2hf(x)| ≤ [NF (f) + ε]ϕ(|h|) for d(x, F ) ≤ 2δ, |h| ≤ δ,

and

(5.33) |f(x)| ≤ A[NF (f) + ε]d(x, F )

3\d(x,F )

ϕ(t)

t2dt for d(x, F ) ≤ 2δ.

SinceT10[ϕ(t)/t2] dt = ∞, there is τ ∈ (0, δ] for which

(5.34)

3\2δ

ϕ(t)

t2dt ≤

2δ\τ

ϕ(t)

t2dt.

By (5.6.3) one can pick a ∈ (0, τ ] such that

(5.35)

3\ϕ(t)

t2dt ≤ τ

2τ\τ

ϕ(t)

t2dt.

We claim that the number thus defined is the one required in (Ext). To show this,

we need to check conditions (i)–(iii) of Proposition 5.2.9 with F = F and with a constant

M ≤ AC, where C := NF (f) + ε.

Estimate (i) is trivially satisfied.

To prove (ii), consider the following two cases.

Case 1: |y − x| ≤ δ. Let L be an interval of length δ such that x, y ∈ L ⊂ [−1, 1].

Since ≤ δ, we use (5.33) and (5.6.1) to obtain

(5.36) ‖f‖L ≤ ACδ

3\2δ

ϕ(t)

t2dt.

Now invoking inequality (5.10) and taking account of (5.32) and (5.36) we have

∣∣∣∣f(y) − f(x)

y − x

∣∣∣∣ ≤ A

[ δ\|y−x|

ω2(f ;L; t)

t2dt+

‖f‖L

δ

]

≤ AC

[ δ\|y−x|

ϕ(t)

t2dt+

3\δ

ϕ(t)

t2dt

]= AC

3\|y−x|

ϕ(t)

t2dt.

Case 2: |y − x| > δ. In this case, using (5.33) and (5.6.1) we argue as follows:

|f(y) − f(x)| ≤ |f(y)| + |f(x)| ≤ AC

3\ϕ(t)

t2dt ≤ AC|y − x|

3\|y−x|

ϕ(t)

t2dt.

Checking condition (iii) splits into the following three cases.

50 L. G. Hanin

Case 1: |z−x| ≤ δ. Let L be an interval of length ≤ 2δ such that x, y, z ∈ L ⊂ [−1, 1].

We apply Proposition 5.2.8 and (5.22) (see also Remark 1 to Proposition 5.2.9) to obtain

∣∣∣∣f(z) − f(x)

z − x−f(y) − f(x)

y − x

∣∣∣∣ ≤ AC

2|z−x|\|y−x|

ϕ(t)

t2dt.

Case 2: |z−x| > δ, |y−x| > τ. Using consecutively (5.33), (5.6.1), (5.35), and finally

(5.6.2) we have

|f(y) − f(x)| ≤ |f(y)| + |f(x)| ≤ AC

3\ϕ(t)

t2dt ≤ ACτ

2τ\τ

ϕ(t)

t2dt

≤ AC|y − x|

2|y−x|\|y−x|

ϕ(t)

t2dt ≤ AC|y − x|

2|z−x|\|y−x|

ϕ(t)

t2dt,

and similarly,

|f(z) − f(x)| ≤ AC|z − x|

2|z−x|\|z−x|

ϕ(t)

t2dt ≤ AC|z − x|

2|z−x|\|y−x|

ϕ(t)

t2dt,

which implies (iii).

Case 3: |z− x| > δ, |y− x| ≤ τ. Using condition (ii) established above and (5.34) we

find that∣∣∣∣f(y) − f(x)

y − x

∣∣∣∣ ≤ AC

3\|y−x|

ϕ(t)

t2dt = AC

[ 2δ\|y−x|

ϕ(t)

t2dt+

3\2δ

ϕ(t)

t2dt

](5.37)

≤ AC

[ 2δ\|y−x|

ϕ(t)

t2dt+

2δ\τ

ϕ(t)

t2dt

]

≤ AC

2δ\|y−x|

ϕ(t)

t2dt ≤ AC

2|z−x|\|y−x|

ϕ(t)

t2dt.

Also, as shown in Case 2,

∣∣∣∣f(z) − f(x)

z − x

∣∣∣∣ ≤ AC

2|z−x|\|y−x|

ϕ(t)

t2dt.

Thus, conditions (i)–(iii) of Proposition 5.2.9 are satisfied, and property (Ext) for

n = 1 follows.

5.6.2. n = 2. Let F be a subset of [−1, 1]2 not lying in a line and f be a function

in Λϕ vanishing on F. Set A(F ) := d(F )/S(F ). For a given ε > 0, we choose δ > 0 for

which ‖f‖F2δ≤ ε,

(5.38) |∆2hf(x)| ≤ [NF (f) + ε]ϕ(|h|) for d(x, F ) ≤ 4δ, |h| ≤ 2δ,

and condition (5.33) is met (all distances are taken in the norm | · |). Next, we pick


τ ∈ (0, δ] subject to (5.34) and ∈ (0, τ ] satisfying (5.35) so that

(5.39)d(F)

S(F)≤ 2A(F ).

We only have to show that, for the function f and the set F, condition (ii) of Pro-

position 5.2.11 is satisfied with a constant ≤ AC, where C := NF (f) + ε. For, if so,

then applying the sufficiency part of Proposition 5.2.11, (5.26), and (5.39) we will find

a function g ∈ Λϕ extending f from F and such that ‖g‖Λϕ≤ A · A(F )[NF (f) + ε], as

required in (Ext).

Let u = (x, y, z) and u′ = (x′, y′, z′) be two triples of points in F, neither lying in a

line, and such that 0 < |y − x| ≤ |z − x|, 0 < |y′ − x′| ≤ |z′ − x′|, and |y − x| ≤ |y′ − x′|.

We will show that

(5.40) |tu − tu′ | ≤ AC

[1

sinα

2|z−x|\|y−x|

ϕ(t)

t2dt+

1

sinα′

2|z′−x′|\|y′−x′|

ϕ(t)

t2dt+

2|y′−x′|+d\|y−x|

ϕ(t)

t2dt

],

where α := αu, α′ := αu′ , d := max|x′ − x|, |y′ − y|, and

(5.41) tu :=f(y) − f(x)

‖y − x‖exy +

1

sinα

[f(z) − f(x)

‖z − x‖−f(y) − f(x)

‖y − x‖cosα

]e⊥xy;

see Proposition 5.2.11 for further notation.

Denote by R the right-hand side of (5.40). When proving (5.40) we consider the

following cases.

Case 1: |z−x| ≤ δ, |z′−x′| ≤ δ, and |x′−x| ≤ 2δ. In this case x, y, z, x′, y′, z′ ⊂

Q(x, 3δ). Applying to the cube Q(x, 3δ) ⊂ F4δ the sufficiency part of Proposition 5.2.11

and the Remark to it, and taking account of (5.38) we obtain (5.40).

Case 2: |z − x| ≤ δ, |z′ − x′| ≤ δ, and |x′ − x| > 2δ. If |y − x| > τ then, as shown in

the case n = 1,

(5.42)|f(y) − f(x)|

‖y − x‖+

|f(z) − f(x)|

‖z − x‖≤ AC

2|z−x|\|y−x|

ϕ(t)

t2dt.

Therefore (see (5.41)), |tu| ≤ R. Since |y′ − x′| ≥ |y − x|, the same argument yields

|tu′ | ≤ R, and (5.40) follows.

Suppose now that |y − x| ≤ τ. Take a point y such that y − x is orthogonal to z − x

and |y − x| = |y − x|. As already shown in Case 1,

|txyz − txyz| ≤ AC

[1

sinα

2|z−x|\|y−x|

ϕ(t)

t2dt+

2|z−x|\|y−x|

ϕ(t)

t2dt+

4|y−x|\|y−x|

ϕ(t)

t2dt

](5.43)

≤AC

sinα

2|z−x|\|y−x|

ϕ(t)

t2dt ≤ R.

Further, recalling that |y−x| ≥ τ and d ≥ 2δ we have, owing to (5.41), (5.33), and (5.37),

52 L. G. Hanin

|txyz| ≤|f(y) − f(x)|

‖y − x‖+

|f(z) − f(x)|

‖y − x‖≤ AC

3\|y−x|

ϕ(t)

t2dt(5.44)

≤ AC

2δ\|y−x|

ϕ(t)

t2dt ≤ AC

2|y′−x′|+d\|y−x|

ϕ(t)

t2dt ≤ R.

From (5.43) and (5.44) we conclude that |tu| ≤ R. Now, depending on whether |y′−x′| is

≤ τ or > τ we apply to u′ either (5.43)–(5.44) or (5.42) to have in both cases |tu′ | ≤ R.

Thus, in Case 2, (5.40) holds.

Case 3: |z − x| ≤ δ, |z′ − x′| > δ. If |y − x| > τ then, as shown in (5.42), |tu| ≤ R.

Let now |y − x| ≤ τ. Following (5.44) we have

|txyz | ≤ AC

2δ\|y−x|

ϕ(t)

t2dt = AC

[ |y′−x′|\|y−x|

ϕ(t)

t2dt+

2δ\|y′−x′|

ϕ(t)

t2dt

]

≤ AC

[ 2|y′−x′|+d\|y−x|

ϕ(t)

t2dt+

2|z′−x′|\|y′−x′|

ϕ(t)

t2dt

]≤ R,

and together with (5.43) this yields |tu| ≤ R.

In like manner, if |y′ − x′| > τ then we know already that |tu′ | ≤ R, while in the case

|y′ − x′| ≤ τ the same conclusion can be drawn by applying (5.43) to the triple u′ and

upon observing that, according to (5.44),

(5.45) |tx′y′z′ | ≤ AC

2δ\|y′−x′|

ϕ(t)

t2dt ≤ AC

2|z′−x′|\|y′−x′|

ϕ(t)

t2dt ≤ R.

Case 4: |z − x| > δ, |z′ − x′| ≤ δ. This case is symmetric to Case 3.

Case 5: |z − x| > δ, |z′ − x′| > δ. If |y − x| > τ then |tu| ≤ R. Now, if |y − x| ≤ τ

then we obtain the same estimate by combining (5.43) with (5.45) applied to the triple

u. In fact, |tu′ | ≤ R is also true.

Thus, in all cases relation (5.40) is valid.

The proof of Theorem 5.1 for n = 2 is now complete.

Appendix

The purpose of the Appendix is to provide the reader with complete self-contained

proofs of Propositions 5.2.2, 5.2.10, 5.2.11, and of relation (5.17).

We proceed from a very general but nonconstructive description of traces of the ge-

neralized Lipschitz spaces Λkϕ, k ≥ 2, determined by differences of order k, on arbitrary

closed sets F ⊂ Q0 (see Theorem A1 in Section 1 below). This description involves

polynomials of degree ≤ k− 1 associated with closed cubes centered at F. For power ma-

jorants, this description appeared in [27], and for general majorants in the “seminormed”

setting in [6]. In our proof of the “normed” version we follow the approach of [6] with

some simplifications. Proposition 5.2.2 is obtained as a simple corollary to Theorem A1.


In the subsequent discussion, we confine ourselves to the Zygmund spaces Λϕ (k = 2).

In this case the above collection of cubes can be reduced to that consisting of cubes with

a point of the set F on the boundary. A process of such reduction was suggested in [45].

The same can be achieved by means of simple explicit formulas; see our proof of Theorem

A1 and also [21]. This leads us to a proof of Proposition 5.2.10. Next, we reformulate

condition (ii)(d) of Proposition 5.2.10 (see Proposition A2.1 below) to obtain (5.17). The

case of two variables is treated in Section 3 where the method of [45] is used to extend the

trace description to the “normed” case with the dependence of the norm of the extension

operator on geometric properties of the set F ⊂ [−1, 1]2 required in Proposition 5.2.11.

1. Traces of generalized Lipschitz spaces. We introduce the following notation

(compare with Section 5.1).

For an integer k and for a function f on Rn, we define the kth difference at a point x

with step h by ∆khf(x) :=

∑ki=0(−1)k−i

(ki

)f(x + ih). The kth modulus of continuity of

f on a cube Q is

ωk(f ;Q; t) := sup|∆khf(x)| : x, x+ kh ∈ Q, |h| ≤ t, t ≥ 0.

Recall that |x| := max1≤i≤n |xi| and note that all distances in the sequel, if not stated

otherwise, are taken in this norm.

Denote by Pk−1 the set of all polynomials in n variables of degree ≤ k − 1. For a

set F in Rn and for a bounded function f on F, we define the best uniform polynomial

approximation of order k − 1 by

Ek−1(f ;Q) := inf‖f − P‖F : P ∈ Pk−1.

A polynomial for which this infimum is attained will be denoted by Pk−1(f ;F ). As shown

in [5],

(A1) Ek−1(f ;Q) ≤ Aωk(f ;Q; dQ)

(the reverse inequality ωk(f ;Q; dQ) ≤ 2kEk−1(f ;Q) trivially holds). Similarly, if Q′ is a

cube centered at a point of Q then

(A2) E1(f ;Q′ ∩Q) ≤ Aω2(f ;Q′ ∩Q; dQ′).

We will need two elementary properties of polynomials. The notation Qt, where Q=

Q(c, d) and t > 0, will be used hereafter for the cube Q(c, td).

Proposition A1.1. For P ∈ Pm and for a cube Q = Q(c, d), we have

(A3) ‖P‖Qt ≤ A(m,n)tm‖P‖Q, t ≥ 1.

Also, if Q′ = Q(c′, d′), where c′ ∈ Q and d′ ≤ λd, then

(A4) ‖P‖Q′ ≤ A(m,n, λ)‖P‖Q′∩Q.

Proof. We check (A3) by induction on m. For m = 0 this is trivial. Let now m ≥ 1

and suppose that (A3) holds for all P ∈ Pm−1. For P ∈ Pm, we have

P (u) = P (c) +

n∑

i=1

(ui − ci)

1\0

∂

∂xiP (c+ s(u− c)) ds.

54 L. G. Hanin

This implies, via the induction hypothesis, that

‖P‖Qt ≤ |P (c)| + ntd max1≤i≤n

∥∥∥∥∂

∂xiP

∥∥∥∥Qt

≤ ‖P‖Q + nA(m− 1, n)tmd max1≤i≤n

∥∥∥∥∂

∂xiP

∥∥∥∥Q

,

and (A3) follows by using the Markov inequality ‖ ∂∂xi

P‖Q ≤ md−1‖P‖Q.

To show (A4), consider the following two cases.

Case (a). If d′ > 2d then Q ⊂ Q′ ⊂ Q(c, (λ+ 1)d). Therefore, by (A3),

‖P‖Q′ ≤ ‖P‖Q(c,(λ+1)d) ≤ A(m,n)(λ + 1)m‖P‖Q = A(m,n, λ)‖P‖Q′∩Q.

Case (b). Let d′ ≤ 2d. Then there is a cube Q′′ = Q(c′′, d′′) with vertex c′ and with

d′′ := mind, d′ such that Q′′ ⊂ Q′ ∩Q. It is easily seen that Q′ ⊂ Q(c′′, 5d′′), hence, by

(A3),

‖P‖Q′ ≤ ‖P‖Q(c′′,5d′′) ≤ A(m,n)‖P‖Q′′ ≤ A(m,n)‖P‖Q′∩Q.

Thus, in both cases (A4) holds. This proves Proposition A1.1.

We define the space Λkϕ := Λk

ϕ(Q0) as the set of all bounded functions on Q0 :=

[−1, 1]n satisfying for some constant M ≥ 0 and for all admissible x and h the following

generalized Lipschitz condition: |∆khf(x)| ≤Mϕ(|h|). Here ϕ is any given nondecreasing

function on R+ such that ϕ(0) = ϕ(0+) = 0 and ϕ(t) > 0 for t > 0. Also, we will

assume without loss of generality that ϕ(t) = 1 for t ≥ 1 and that the function ϕ(t)/tk

is nonincreasing. The latter implies that

ϕ(λt) ≤ λkϕ(t), λ ≥ 1.

This inequality will be systematically used in the sequel.

The space Λkϕ is endowed with the norm ‖f‖k,ϕ := max‖f‖Q0

, |f |k,ϕ,where |f |k,ϕ :=

inf M over all constants M involved in the definition of the space Λkϕ.

If F is a closed subset of Q0 then we define in a usual way the trace space Λkϕ(F ) :=

f |F : f ∈ Λkϕ, and supply it with the norm

‖f‖k,ϕ;F := inf‖f‖k,ϕ : f |F = f, f ∈ Λkϕ(F ).

For a closed set F ⊂ Q0, we denote by KF the collection of all cubes Q(c, d) with

c ∈ F and d ≤ A0, where A0 is an absolute constant (we may take, for example, A0 = 66).

Throughout the Appendix, the letter A (with or without an index) stands for a

positive constant which may depend only on n and k.

Theorem A1. Let F be a closed subset of Q0 and f be a function on F. Then

f ∈ Λkϕ(F ) iff there is a mapping that associates with every cube Q ∈ KF a polynomial

PQ ∈ Pk−1 such that for some constant M ≥ 0 the following conditions are satisfied :

(i) PQ(cQ) = f(cQ) for all Q ∈ KF ;

(ii) ‖PQ‖Q ≤M for all Q ∈ KF ;

(iii) ‖PQ′ − PQ‖Q′ ≤Mϕ(dQ) for all Q′, Q ∈ KF such that Q′ ⊂ Q.

Also,

A1‖f‖k,ϕ;F ≤ inf M ≤ A2‖f‖k,ϕ;F , f ∈ Λkϕ(F ).


Proof. Necessity. Suppose that f ∈ Λkϕ(F ) and put C := 2‖f‖k,ϕ;F . There exists

a function f ∈ Λkϕ such that f |F = f and ‖f‖k,ϕ ≤ C. For Q ∈ KF , set P 0

Q :=

Pk−1(f ;Q∩Q0) and PQ := P 0Q−P 0

Q(cQ)+f(cQ). We claim that the collection PQQ∈KF

of polynomials meets conditions (i)–(iii) with M ≤ AC.

First, condition (i) is obviously satisfied. Next, we check (ii). By Proposition A1.1 we

have ‖PQ‖Q ≤ A‖PQ‖Q∩Q0. Combining this with

‖PQ‖Q∩Q0≤ 2‖P 0

Q‖Q∩Q0+ |f(cQ)| ≤ 2‖P 0

Q − f‖Q∩Q0+ 3‖f‖Q∩Q0

≤ 5‖f‖Q∩Q0,

we obtain ‖PQ‖Q ≤ AC. Passing to (iii), take a pair of cubes Q′, Q ∈ KF with Q′ ⊂ Q.

Invoking (A4) and (A2) we have

‖PQ′ − PQ‖Q′ ≤ ‖P 0Q′ − P 0

Q‖Q′ + |P 0Q′(c′Q) − f(cQ′)| + |P 0

Q(cQ) − f(cQ)|

≤ A‖P 0Q′ − P 0

Q‖Q′∩Q0+ ‖P 0

Q′ − f‖Q′∩Q0+ ‖P 0

Q − f‖Q∩Q0

≤ A(‖P 0Q′ − f‖Q′∩Q0

+ ‖P 0Q − f‖Q∩Q0

)

≤ A(‖P 0Q − f‖Q′∩Q0

+ ‖P 0Q − f‖Q∩Q0

)

≤ A‖P 0Q − f‖Q∩Q0

≤ Aωk(f ;Q ∩Q0; dQ) ≤ ACϕ(dQ).

Thus, conditions (i)–(iii) are satisfied with a constant M ≤ A‖f‖k,ϕ;F .

Sufficiency. Let WF be a Whitney decomposition of Rn \ F into a family of closed

cubes with disjoint interiors. An argument similar to that in [48], Chapter 6, Section 1,

shows that the cubes constituting WF can be chosen to meet the following requirements:

(a) 2dQ ≤ dist(Q,F ) ≤ 8dQ for all Q ∈WF ;

(b) if Q1, Q2 ∈ WF and Q1 ∩Q2 6= ∅ then 1/4 ≤ dQ1/dQ2

≤ 4;

(c) Qt ∩ F = ∅ for every Q ∈WF and 0 < t < 3;

(d) if Q1, Q2 ∈WF and Q1 ∩Q2 = ∅ then for 1 < t < 4/3 we have Qt1 ∩Q

t2 = ∅;

(e) the covering WF is locally finite with multiplicity < 6n.

For Q ∈ WF , define Q∗ := Q4/3, and construct a smooth partition of unity by C∞-

functions ψQQ∈WFsuch that ψQ ≥ 0, suppψQ ⊂ Q∗, and ‖DαψQ‖∞ ≤ Ad

−|α|Q for

1 ≤ |α| ≤ k.

For Q ∈ WF , define Q := Q(aQ, dQ), where aQ is a point of F closest to Q (in the

norm | · |). The Whitney extension of the function f is defined by f(x) = f(x) for x ∈ F

and

(A5) f(x) =∑

Q∈WF

PQ(x)ψQ(x) for x ∈ Rn \ F.

We seek to prove that f ∈ Λkϕ and ‖f‖k,ϕ ≤ AM.

First, note that the sum in (A5) may be restricted to the subfamily W ′F of cubes

Q ∈WF for which Q∗ intersects Q0, and a simple geometric argument based on property

(a) shows that for every such Q ∈ W ′F we have dQ ≤ 6/5.

Further, it is easily seen that for every cube Q ∈ WF , Q∗ ⊂ QA1 with A1 = 31/3.

Hence, taking into account inequality (A3) and condition (ii) we have, for every Q ∈W ′F ,

‖PQ‖Q∗ ≤ ‖PQ‖QA1≤ A‖PQ‖Q ≤ AM.

56 L. G. Hanin

This implies ‖f‖Q0\F ≤ AM. Besides this, owing to (i) and (ii) we have ‖f‖F ≤ M.

Therefore, ‖f‖Q0≤ AM.

Now we will show that |f |k,ϕ ≤ AM. According to (A1) it is sufficient to check that

Ek−1(f ;K) ≤ AMϕ(dK) for every cube K ⊂ Q0.

Suppose first that cK ∈ F. For every cube Q ∈ WF , we have dist(Q∗, F ) ≥ (5/3)dQ.

Hence, if Q∗ ∩ K 6= ∅ then dQ ≤ (3/5)dK . Using simple geometric considerations we

derive from this that Q ⊂ KA2 with A2 = 39/5. Now set K := KA2 . We claim that

‖f − PK‖K ≤ AMϕ(dK).

For x ∈ F ∩ K, we put Q′ = x and Q = K in (iii) to obtain |f(x) − PK

(x)| ≤

Mϕ(dK

) ≤ AMϕ(dK).

Let now x ∈ K \ F. We have

(A6) f(x) − PK

(x) =∑

Q∈W ′

F

[PQ(x) − PK

(x)]ψQ(x).

For every cube Q ∈W ′F with Q∗ ∩K 6= ∅, we find using (A3) and (iii) that

‖PQ − PK‖Q∗ ≤ ‖PQ − P

K‖QA1

≤ A‖PQ − PK‖Q ≤ AMϕ(dK).

Therefore, we conclude from (A6) that ‖f − PK‖K\F ≤ AMϕ(dK). Thus, in the case

cK ∈ F,

(A7) Ek−1(f ;K) ≤ ‖f − PK‖K ≤ AMϕ(dK).

Suppose now that cK 6∈ F and fix a cube Q ∈ W ′F containing the point cK . For the

cube Q, we have two possibilities.

Case 1: K is not contained in Q∗. In this case we obviously have dK ≥ dQ/3. Also,

using property (a) it is easy to see that K ⊂ KA3 with A3 = 32. We set K ′ := KA3 and

upon observing that cK′ ∈ F we make use of (A7) to obtain

Ek−1(f ;K) ≤ Ek−1(f ;K ′) ≤ AMϕ(d′K) ≤ AMϕ(dK).

Case 2: K ⊂ Q∗. For a polynomial P ∈ Pk−1 (that will be chosen later) we put

(A8) g(x) := f(x) − P (x) =∑

Q′∈W ′

F

[PQ′(x) − P (x)]ψQ′ (x), x ∈ Rn \ F.

Let

T (x) :=∑

|α|≤k−1

Dαg(cK)(x− cK)α

α!

be the Taylor polynomial for the function g of order k − 1 at the point cK (note that

f ∈ C∞(Rn \ F )). A standard estimate for the Taylor remainder gives

(A9) ‖g − T ‖K ≤ AdkK max

|α|=k‖Dαg‖Q∗ .

Using properties (d) and (e) of the collection WF we find that for every x ∈ int Q∗, the

sum in (A8) ranges over < 6n cubes Q′ ∈ W ′F for which Q′ ∩ Q 6= ∅. For every such

cube Q′ and for |α| = k, we have, using the Leibniz formula, the Markov inequality, and

property (c) of the Whitney cubes,


‖Dα[(PQ′ − P )ψQ′ ]‖Q∗ ≤ Amaxβ≤α

d|β|−kQ′ ‖Dβ(PQ′ − P )‖Q∗

≤ Ad−|β|Q∗ d

|β|−kQ′ ‖PQ′

− P‖Q∗ ≤ Ad−kQ ‖PQ′

− P‖Q.

Hence, in view of (A8),

(A10) ‖Dαg‖Q∗ ≤ Ad−kQ max

Q′∩Q6=∅‖PQ′

− P‖Q, |α| = k.

Note that for any two cubes Q,Q′ ∈ WF with Q ∩ Q′ 6= ∅ we have Q ⊂ (Q′)A4 and

Q′ ⊂ QA5 , where one can take A4 = 18 and A5 = 54. We define P := PQA5to obtain, by

Proposition A1 and in view of condition (iii),

‖PQ′ − P‖Q ≤ ‖PQ′ − P‖(Q′)A4(A11)

≤ A‖PQ′− PQA5

‖Q′≤ AMϕ(A5dQ) ≤ AMϕ(dQ).

Hence by (A10) we have ‖Dαg‖Q∗ ≤ Ad−kQ ϕ(dQ). Therefore, recalling that g = f − P

and taking account of (A9) we obtain the estimate

Ek−1(f ;K) ≤ ‖f − P − T ‖K = ‖g − T ‖K ≤ AM(dK/dQ)kϕ(dQ).

In our case, dK ≤ dQ∗ = (4/3)dQ, hence ϕ(dQ)/dkQ ≤ Aϕ(dK)/dk

K . Continuing (A11) we

finally have

(A12) Ek−1(f ;K) ≤ AMϕ(dK).

Observe that for an appropriate choice of the constant A0 in the definition of the

family KF all cubes involved in the proof belong to KF , and therefore (A12) holds for

every cube K ⊂ Q0. Hence, owing to (A1), |f |k,ϕ ≤ AM.

Thus, f ∈ Λkϕ(F ) and ‖f‖k,ϕ;F ≤ AM, as required.

Remark. Theorem A1 is valid for any cube Q in place of Q0 with KF defined to

consist of all cubes Q(c, d) with c ∈ F and d ≤ A0dQ. Furthermore, Theorem A1 is also

true for Q = Rn and thus provides a description of the trace of the space Λk

ϕ(Rn) to any

closed subset F in Rn. If F is bounded then every function f ∈ Λk

ϕ(F ) can be extended

to a function in Λkϕ(Rn) with compact support.

Proposition A1.2. Let Q = Q(c, d) be a cube in Rn. Every bounded function f on

Q can be extended to a function f on Q := Q(c, d) with d ≤ Ad such that

‖f‖Q ≤ A‖f‖Q and ωk(f ; Q; t) ≤ Aωk(f ;Q; t), t ≥ 0.

Proof. Consider the space Λkϕ(Q) with ϕ := ωk(f ;Q). Define KQ as in the remark

to Theorem A1. For K ∈ KQ, let

PK := Pk−1(f ;K ∩Q) − Pk−1(f ;K ∩Q)(cK) + f(cK).

A word-for-word repetition of the proof of the necessity part of Theorem A1 shows that

these polynomials satisfy conditions (i), (ii) with a constant M1 ≤ A‖f‖Q, and also

condition (iii) with a constant M2 ≤ A. Now applying the sufficiency part of Theorem

A1 and observing that the extension f satisfies ‖f‖Q ≤ AM1 and |f |k,ϕ ≤ AM2 we obtain

the required conclusion.

Proposition 5.2.2 is a particular case of Proposition A1.2 with k = 2.

58 L. G. Hanin

2. Traces of Zygmund spaces. In the case k = 2, a more constructive version of

Theorem A1 is possible. This is Proposition 5.2.10, which we are going to prove now.

Proof of Proposition 5.2.10. Necessity. Set M := ‖f‖k,ϕ. Then, for such M,

condition (i) is obviously satisfied. For x, y ∈ F, x 6= y, define txy := ∇Pxy, where Pxy is

the interpolation polynomial described in Proposition 5.2.8. According to the Corollary

to that proposition, conditions (a)–(d) in (ii) are also met.

Sufficiency. Suppose a function f on F meets conditions (i) and (ii). By Theorem

A1 we need to show that for every cube Q = Q(c, d) ∈ KF there exists a polynomial

PQ(u) = f(c) + tQ(u− c) that satisfies conditions (ii) and (iii) of Theorem A1.

We introduce a few geometric characteristics of the mutual position of the sets F and

Q. Fix a point aQ ∈ Q ∩ F, put rQ := maxx∈Q∩F |x − aQ|, and denote by xQ a point in

Q ∩ F for which |xQ − aQ| = rQ. Clearly,

(A13.1) 12 diam(Q ∩ F ) ≤ rQ ≤ diam(Q ∩ F ) ≤ 2.

If F is not contained in Q then define also Q := maxrQ, infy∈F\Q |y−aQ|, and denote

by yQ a point in F for which Q = |yQ − aQ|. Observe that

(A13.2) rQ ≤ Q ≤ diamF ≤ 2.

We define the required vector field tQ, Q ∈ KF , as follows:

Case 1. If d = 0 then tQ := 0.

Case 2. If d 6= 0 but Q ∩ F = c then tQ := tcyQ.

Case 3. If Q ∩ F 6= c and Q ∩ F 6= F then tQ := θQtaQxQ+ (1 − θQ)taQyQ

, where

θQ :=T2Q

dQ[ϕ(t)/t2] dt/

T2Q

rQ[ϕ(t)/t2] dt.

Case 4. If F ⊂ Q then tQ := γQtaQxQ, where γQ :=

T3dQ

[ϕ(t)/t2] dt/T3rQ

[ϕ(t)/t2] dt.

In the sequel, the magnitudes defined above will be written without the index Q,

while those related to a cube Q′ will be marked with the symbol “′”.

For the sake of simplicity we will use the notation Φ(a, b) :=Tba[ϕ(t)/t2] dt. Note the

following obvious properties of the function Φ, for our further reference:

Φ(a, b) ≤ a−1φ(b), 0 < a ≤ b;(A14.1)

|Φ(a, b)| ≤ Φ(mina, b,maxa, b);(A14.2)

if 0 < a2 ≤ a1 ≤ b1 ≤ b2 then Φ(a1, b1) ≤ Φ(a2, b2).(A14.3)

We will check condition (ii) of Theorem A1. Since ‖P‖Q ≤ |f(c)|+Ad|t| and |f(c)| ≤

M, we only have to show that d|t| ≤ AM.

In Case 1, t = 0. Next, in Case 2, r = 0 and = |y − c| ≥ d, hence by (ii)(c) and

(A14.1) we have d|t| ≤MdΦ(|y − c|, 3) ≤M.

Now consider Case 3. Note that in this case ≥ d/3. For, if diam(F ∩ Q) < 2d/3

then for all z ∈ F \ Q, |z − a| ≥ |z − c| − |c − a| ≥ d − 2d/3 = d/3. Otherwise, i.e. for

diam(F ∩Q) ≥ 2d/3, by (A13.1) and (A13.2) we again have ≥ diam(F ∩Q)/2 ≥ d/3.

Taking into account condition (ii)(c), (A14.1), and the above definitions of t := tQ we


find that, in Case 3,

d|t| ≤ d(|tay| + |θ| · |tax − tay|) ≤Md[Φ(, 3) + |Φ(d, 2)|]

≤M [d/+ dΦ(2d/3, 4)] ≤ AM.

Finally, in Case 4, by (A14.1) we have d|t| ≤Md|Φ(d, 3)| ≤ AM.

We now pass to checking condition (iii) of Theorem A1. We write

(PQ − PQ′)(u) = [f(c′) − f(c) − t(c′ − c)] + (t′ − t)(u − c′),

therefore

‖PQ − PQ′‖Q′ ≤ |f(c′) − f(c) − t(c′ − c)| +Ad′|t′ − t| := I1 +AI2.

Our first step consists in estimating I1 := |f(c′) − f(c) − t(c′ − c)|.

In Cases 1 and 2, c′ = c, hence I1 = 0. While considering cases 3 and 4 one may

assume that c′ 6= c. We claim that

(A15) |c′ − c| · |tcc′ − tax| ≤ AMϕ(d).

To show this, observe that in view of (A13.1), |x− a| = r ≥ |c′ − c|/2. Hence if |x− c| ≥

|c′ − c|/4 then due to (ii)(b), (ii)(d), and to relations (A14),

|tcc′ − tax| ≤ |tcc′ − tcx| + |tcx − tax| ≤MΦ(|c′ − c|/4, 2d) ≤ AMϕ(d)

|c′ − c|.

Else, we have |a− c| ≥ |c′ − c|/4, and for similar reasons

|tcc′ − tax| ≤ |tcc′ − tca| + |tac − tax| ≤ AMϕ(d)

|c′ − c|,

which completes the proof of (A15).

By (ii)(a) we have f(c′) − f(c) − t(c′ − c) = (tcc′ − t)(c′ − c), therefore in Case 3 we

make use of (ii)(b), (ii)(d), (A14), and (A15) to find that

I1 ≤ A|tcc′ − t| · |c′ − c| ≤ A|c′ − c| · [|tcc′ − tax| + |1 − θ| · |tax − tay|]

≤ AM [ϕ(d) + |c′ − c| · |Φ(r, d)|] ≤ AM [ϕ(d) + |c′ − c|Φ(|c′ − c|/2, 2d)] ≤ AMϕ(d).

In like manner, in Case 4,

I1 ≤ A|c′ − c| · [|tcc′ − tax| + |1 − γ| · |tax|] ≤ AM [ϕ(d) + |c′ − c| · |Φ(r, d)|] ≤ AMϕ(d).

When estimating I2 := d′|t′ − t| we may assume that d′ > 0. If F ∩ Q = c then

t = t′, i.e. I2 = 0. In the case F ∩Q′ = c′ we have

d′ ≤ |y′ − c′| ≤ diam(F ∩Q) ≤ 2|x− a|,

hence repeating the above argument with y′ in place of c we obtain the estimate

I2 = d′|t′ − t| ≤ |y′ − c′| · |ty′c′ − t| ≤ AMϕ(d).

Thus we are left with the combinations of Cases 3 and 4 for the cubes Q and Q′.

If both of them fall into Case 3 and F ∩Q′ = F ∩Q then a′ = a, x′ = x, y′ = y, r′ = r,

and ′ = . Therefore, recalling our definitions of tQ and invoking (ii)(d) and (A13.1) we

have

I2 = d′|θ′ − θ| · |tax − tay| ≤Md′Φ(d′, d) ≤Mϕ(d).

60 L. G. Hanin

Next, if F ∩Q′ 6= F ∩Q then

|t′ − t| ≤ |θ′| · |ta′x′ − ta′y′ | + |ta′y′ − tax| + |1 − θ| · |tax − tay| := I21 + I22 + I23.

Since d′/3 ≤ ′ ≤ 2d, one has, using relations (A14),

I21 ≤M |Φ(d′, 2′)| ≤MΦ(2d′/3, 2d) ≤ AMϕ(d)/d′.

Further, in the case under study there is a point z ∈ (F∩Q)\Q′, hence 2r ≥ diam(F∩Q) ≥

|z − c′| ≥ d′. Therefore,

(A16) I23 ≤M |Φ(r, d)| ≤M |Φ(d′/2, 2d)| ≤ AMϕ(d)/d′.

To estimate I22, observe that r = |x − a| ≥ d′/2, hence either |x − a′| ≥ d′/4, in which

case upon recalling that |y′ − a′| = ′ ≥ d′/3 we have

I22 ≤ |ta′y′ − ta′x| + |txa′ − txa| ≤ 2MΦ(d′/4, 2d) ≤ AMϕ(d)/d′,

or |a− a′| ≥ d′/4, and in the latter case we get in quite a similar way the same estimate.

Thus, |t′ − t| ≤ AMϕ(d)/d′, whence I2 ≤ AMϕ(d).

Suppose now that F ∩Q′ pertains to Case 3 and F ∩Q belongs to Case 4. We have

|t′ − t| ≤ |θ′| · |ta′x′ − ta′y′ | + |ta′y′ − ta′x| + |1 − γ| · |tax| := I21 + I22 + I ′23.

The terms I21 and I22 are estimated exactly as above whereas for I ′23 we apply estimates

(A16).

Let finally Q′ fall into Case 4; then the same is necessarily true for Q. This implies

r′ = r, and therefore

I2 = d′|γ′ − γ| · |tax| ≤Md′Φ(d′, d) ≤Mϕ(d).

Scrutinizing the above estimates we conclude that in all cases I1 + I2 ≤ AMϕ(d).

Consequently, condition (iii) of Theorem A1 is satisfied.

By Theorem A1, there exists a function f ∈ Λϕ such that f |F = f and ‖f‖Λϕ≤ AM.

The proof is finished.

Our next objective is to prove estimate (5.17). It will be derived as a corollary to the

following multivariate statement.

For a subset F in Rn, we let hereafter F := (x, y) : x, y ∈ F, x 6= y.

Proposition A2.1. Let F be a subset of Rn, and let txy, (x, y) ∈ F , be a vector field

such that txy = tyx for all (x, y) ∈ F and

(A17) |txy − txz| ≤MΦ(|y − x|, 2|z − x|)

for all x, y, z ∈ F with 0 < |y−x| ≤ |z−x|. Then for all pairs u = (x, y) and u′ = (x′, y′)

in F such that 0 < |y − x| ≤ |y′ − x′|, we have

(A18) |tu − tu′ | ≤ AM(u, u′),

where

(A19) (u, u′) := Φ(|y − x|, 2|y′ − x′| + |u′ − u|)

and |u′ − u| := max|x′ − x|, |y′ − y|.


Proof. Suppose, for example, that |u′−u| = |x′−x|.We have either |x′−x| ≥ |y−x|/2

or |x′ − y| ≥ |y − x|/2. In the first case, using the inequality

(A20) Φ(λa, λb) ≤ maxλ, λ−1Φ(a, b), 0 < a ≤ b, λ > 0,

we find that

|tu − tu′ | ≤ |txy − txx′ | + |tx′x − tx′y′ |

≤MΦ(|y − x|/2, 2 max|y − x|, |x′ − x|)

+MΦ(|y − x|/2, 2 max|y′ − x′|, |x′ − x|)

≤ 2MΦ(|y − x|/2, 2 max|y′ − x′|, |x′ − x|) = 2MΦ(|y − x|/2, |y − x|)

+ 2MΦ(|y − x|, 2|y − x|) + 2MΦ(2|y − x|, 2 max|y′ − x′|, |x′ − x|)

≤ 6MΦ(|y − x|, 2|y − x|) + 4MΦ(|y − x|,max|y′ − x′|, |x′ − x|)

≤ 10MΦ(|y − x|, 2|y′ − x′| + |x′ − x|).

If |x′ − y| ≥ |y − x|/2 then via a similar argument we obtain

|tu − tu′ | ≤ |txy − tx′y| + |tx′y − tx′y′ |

≤ 6MΦ(|y − x|, 2|y − x|) + 4MΦ(|y − x|,max|y′ − x′|, |x′ − y|)

≤ 10MΦ(|y − x|, 2|y′ − x′| + |x′ − x|).

Thus, |tu − tu′ | ≤ 10Mr(u, u′).

Remark 1. Conversely, inequality (A18) which holds with a constant C implies (A17)

with a constant ≤ AC. In fact, if 0 < |y− x| ≤ |z − x| then setting u = (x, y), u′ = (x, z)

in (A18) and (A19), we have

|txy − txz| ≤ CΦ(|y − x|, 2|z − x| + |z − y|)

≤ CΦ(|y − x|, 2|z − x|) + CΦ(2|z − x|, 4|z − x|)

≤ 3CΦ(|y − x|, 2|z − x|).

Thus, for symmetric vector fields, conditions (A17) and (A18) are equivalent.

Remark 2. Let n = 1, F = [−1, 1], and f ∈ Λϕ. By the Corollary to Proposition

5.2.8, txy = (f(y)− f(x))/(y−x) satisfies condition (A17). Therefore, (A18) is also true.

This justifies (5.17).

Observe that (A18) is a Lipschitz condition with respect to the function on F defined

by (A17), (u′, u) := (u, u′) if u′ 6= u, and (u, u′) := 0 if u′ = u.

Proposition A2.2. is a metric on F .

Proof. We only need to check the triangle inequality. Let ui = (xi, yi), i = 1, 2, 3,

be three points in F . We will show that

(A21) (u1, u3) ≤ (u1, u2) + (u2, u3).

Set ri = |yi−xi|, i = 1, 2, 3.We may assume that r1 ≤ r3. Define d1 := |u2−u1|, d2 :=

|u3 − u1|, and d3 := |u3 − u2|; then d2 − d1 ≤ d3 ≤ d1 + d2. Consider the following cases:

Case 1: r2 ≤ r1. If d2 ≤ d3 then (A21) follows from the estimate

(u1, u3) = Φ(r1, 2r3 + d2) ≤ Φ(r2, 2r3 + d3) = (u2, u3).

62 L. G. Hanin

Let now d2 > d3. Then by (A14.3) we have

(u1, u3) = Φ(r1, 2r3 + d3) + Φ(2r3 + d3, 2r3 + d2)

≤ Φ(r2, 2r3 + d3) + Φ(r2, 2r1 + d2) = (u2, u3) + (u1, u2).

Case 2: r1 < r2 ≤ r3. If 2r3 + d2 ≤ 2r2 + d1 then

(u1, u3) ≤ Φ(r1, 2r2 + d1) = (u1, u2),

which implies (A21), whereas in the opposite case we have, by (A14.3),

(u1, u3) = Φ(r1, 2r2 + d1) + Φ(2r2 + d1, 2r3 + d2) ≤ Φ(r1, 2r2 + d1) + Φ(2r2, 2r3 + d3)

≤ Φ(r1, 2r2 + d1) + Φ(r2, 2r3 + d3) = (u1, u2) + (u2, u3).

Case 3: r2 > r3. Again, on the basis of (A14.3),

(u1, u3) = Φ(r1, 2r3 + d2) ≤ Φ(r1, 2r3 + d1 + d3)

= Φ(r1, 2r3 + d1) + Φ(2r3 + d1, 2r3 + d1 + d3)

≤ Φ(r1, 2r2 + d1) + Φ(r3, 2r2 + d3) = (u1, u2) + (u2, u3),

and (A21) follows.

3. Proof of Proposition 5.2.11. We start with recalling the following well-known

result which goes back to the ideas of E. J. McShane [37] and is a ramification of the

Helly theorem; see also [30] and [45].

Proposition A3.1. Let Qα = Q(cα, dα), α ∈ ℵ, be a family of cubes in Rn, and

be a semimetric on ℵ. Suppose that for every α, α′ ∈ ℵ there are points aα,α′ ∈ Qα and

aα′,α ∈ Qα′ such that |aα,α′ − aα′,α| ≤ (α, α′). Then for every α ∈ ℵ, one can choose a

point aα ∈ Qα such that |aα − aα′ | ≤ (α, α′) for all α, α′ ∈ ℵ.

Proof. It suffices to prove the proposition for n = 1. We claim that

aα := infcα′ + dα′ + (α, α′) : α′ ∈ ℵ, α ∈ ℵ,

is the required collection of points. First, aα ≤ cα + dα. Next, for every α′ ∈ ℵ,

cα′ + dα′ + (α, α′) ≥ aα′,α + (α, α′) ≥ aα,α′ ≥ cα − dα,

hence aα ≥ cα − dα. Therefore, aα ∈ Qα for all α. Further, for every β ∈ ℵ,

aα ≤ cβ + dβ + (α, β) ≤ cβ + dβ + (α′, β) + (α, α′).

Taking infimum over β ∈ ℵ we obtain the inequality aα ≤ aα′ + (α, α′), whence for

symmetry reason we conclude that |aα − aα′ | ≤ (α, α′).

Now we are prepared to prove Proposition 5.2.11.

Proof of Proposition 5.2.11.Necessity. SupposeF is a closed subset of [−1, 1]2 and

f ∈ Λϕ(F ). For C := ‖f‖Λϕ(F ), there exists a function in the space Λϕ := Λϕ([−1, 1]2)

whose restriction to the set F is f and the norm in the space Λϕ is not larger than 2C.

For this function, we retain the notation f.

We will show that f satisfies conditions (i) and (ii) of Proposition 5.2.11. The first

of them is trivially valid with M = 2C. Now we prove (ii). For a triple u = x, y, z of

distinct points of F not lying in a line, denote by Pu the (unique) polynomial of degree


≤ 1 that interpolates f at these three points. It is easy to see that Pu(s) = f(x)+tu(s−x),

where tu = txyz is the vector involved in condition (ii) of Proposition 5.2.11.

Let Pxy and Pxz be the polynomials interpolating f at points x, y and x, z, respectively,

as defined in Proposition 5.2.8. We write these polynomials in the form Pxy(s) = f(x) +

txy(s− x), Pxz(s) = f(x) + txz(s − x). Assuming that 0 < |y − x| ≤ |z − x| we have by

Proposition 5.2.8 the following estimate:

‖txy − txz‖ ≤ A

2|z−x|\|y−x|

ω2(f ; t)

t2dt ≤ ACΦ(|y − x|, 2|z − x|).

The lines t(y − x) = f(y) − f(x) and t(z − x) = f(z) − f(x) in R2 form the angle

minαu, π − αu, where αu is the angle between the vectors exy and exz (see condition

(ii) of Proposition 5.2.11), pass through the points txy and txz, respectively, and intersect

at the point txyz. Denote by d the Euclidean distance from the point txy to the line

t(z − x) = f(z) − f(x). Then

‖txy − txyz‖ =d

sinαu≤

‖txy − txz‖

sinαu≤

AC

sinαuΦ(|y − x|, 2|z − x|).

Now using Proposition A2.1 we conclude that for two triples u, u′ of points of F as in

condition (ii) of Proposition 5.2.11,

|tu − tu′ | ≤ |txyz − txy| + |txy − tx′y′ | + |tx′y′ − tx′y′z′ |

≤ AC

[Φ(|y − x|, 2|z − x|)

sinαu+ Φ(|y − x|, 2|y′ − x′| + max|x′ − x|, |y′ − y|)

+Φ(|y′ − x′|, 2|z′ − x′|)

sinαu′

].

Thus, conditions (i) and (ii) hold with a constant M ≤ AC, and this also proves the

right-hand estimate in (5.26).

Sufficiency. Let F be a closed subset of [−1, 1]2 not contained in a line, and let f be

a function on F that meets conditions (i) and (ii) of Proposition 5.2.11 with a constant

M. We may (and will) assume that M = 1. Due to Proposition 5.2.10 we only have to

show that there exists a vector field txy, (x, y) ∈ F , which satisfies conditions (ii)(a)–(d)

of that proposition with a constant ≤ Ad(F )/S(F ) (the reader is reminded that d(F ) is

the Euclidean diameter of F and S(F ) is the area of the convex hull of F ).

Let V be the set of all ordered triples v = (x, y, z) of distinct points of F not lying in a

line. Define a semimetric d on V by d(v, v′) := ((x, y), (x′, y′)), where v = (x, y, z), v′ =

(x′, y′, z′), and is the metric on F defined above. To every v ∈ V we relate the cube

Qv := Q(tv, dv) with dv := r(v)/sinαv, where

r(v) := Φ(min|y − x|, |z − x|, 2 max|y − x|, |z − x|), v = (x, y, z).

In these terms, condition (ii) of Proposition 5.2.11 can be rewritten in the form

|tv − tv′ | ≤ dv + dv′ + d(v, v′), v, v′ ∈ V .

Geometrically, this means that there are points avv′ ∈ Qv and av′v ∈ Qv′ such that

|avv′ − av′v| ≤ d(v, v′). Applying Proposition A3.1 we infer that for every v ∈ V there

64 L. G. Hanin

exists a point av ∈ Qv such that |av − av′ | ≤ d(v, v′) for all v, v′ ∈ V , i.e. that

|axyz − ax′y′z′ | ≤ ((x, y), (x′, y′)).

Setting here x′ = x and y′ = y we see that axyz = axyz′ for all z, z′ ∈ F with

(x, y, z), (x, y, z′) ∈ V , that is, axyz is independent of z and will be denoted in the sequel

by axy. Thus, the previous inequality assumes the form

(A22) |axy − ax′y′ | ≤ ((x, y), (x′, y′)), (x, y), (x′, y′) ∈ F .

In particular, by Remark 1 to Proposition A2.1,

|axy − axz| ≤ Ar(v)

for every v = (x, y, z) ∈ V . Also, (A22) implies that

|ayx − azx| ≤ ((y, x), (z, x)) = ((x, y), (x, z)) ≤ Ar(v).

Since axy ∈ Qv, we have

|axy − txyz| ≤r(v)

sinαv, v = (x, y, z).

Hence, by (A22),

|ayx − txyz| ≤ |ayx − axy| + |axy − txyz| ≤ Φ(|y − x|, 3|y − x|) +r(v)

sinαv

≤ 2Φ(|y − x|, 2|y − x|) +r(v)

sinαv≤

3r(v)

sinαv.

Therefore, setting bxy := (axy + ayx)/2, (x, y) ∈ F , we see that there is a symmetric

vector field b satisfying condition (ii)(d) of Proposition 5.2.10 and such that

(A23) |bxy − txyz| ≤Ar(v)

sinαv, v = (x, y, z).

To comply with condition (ii)(a) of Proposition 5.2.10, we set txy := Prxybxy, (x, y) ∈

F , where Prxy stands for the orthogonal projection onto the line t(y − x) = f(y)− f(x).

Observe that txy, (x, y) ∈ F , is a symmetric vector field satisfying the condition (ii)(a).

We claim that the condition (ii)(d) also holds. In fact, if 0 < |y − x| ≤ |z − x| then

|txy − txz| ≤ ‖txy − txz‖ = ‖Prxybxy − Prxzbxz‖

≤ ‖Prxzbxy − Prxzbxz‖ + ‖Prxybxy − Prxzbxy‖

≤ ‖bxy − bxz‖ + I ≤ AΦ(|y − x|, 2|z − x|) + I,

where I := ‖Prxybxy − Prxzbxy‖. To estimate I, put p := Prxybxy and q := Prxzbxy. A

simple geometric argument together with (A23) shows that

I = ‖p− q‖ ≤ Amax‖bxy − p‖, ‖bxy − q‖ sinαv

≤ A‖bxy − tv‖ sinαv ≤ AΦ(|y − x|, 2|z − x|).

Therefore, for the vector field txy, (x, y) ∈ F , the condition (ii)(d) is met.

Finally, we have to check condition (ii)(b) of Proposition 5.2.10, i.e. that

|txy| ≤ Ad(F )

S(F )Φ(|y − x|, 3), (x, y) ∈ F .


Take points x0, z0 in F such that d(F ) = ‖z0−x0‖. Let H be the height of a rectangle

with base [x0, z0] containing F and having the minimal area, which we denote by S. There

is a point y0 ∈ F whose Euclidean distance from the interval [x0, z0] is not less than H/2.

We may assume in fact that ‖y0−x0‖ ≥ d(F )/2. Let α0 be the angle between the vectors

y0 − x0 and z0 − x0. Then

2‖z0 − x0‖ ‖y0 − x0‖ sinα0 ≥ S ≥ S(F ),

hence sinα0 ≥ S(F )/[2d2(F )].

Note also that both |f(y0) − f(x0)|/‖y0 − x0‖ and |f(z0) − f(x0)|/‖z0 − x0‖ do not

exceed A/d(F ). Therefore, upon observing that S(F ) ≤ d2(F ), we have for

tv0=f(y0) − f(x0)

‖y0 − x0‖ex0y0

+1

sinα0

[f(y0) − f(x0)

‖y0 − x0‖−f(z0) − f(x0)

‖z0 − x0‖cosα0

]e⊥x0y0

with v0 := (x0, y0, z0) the estimate |tv0| ≤ Ad(F )/S(F ). Next, in view of (A23) and

(A14.1) we conclude that

|tx0z0− tv0

| ≤ ‖tx0z0− tv0

‖ ≤ ‖bx0z0− tv0

‖ ≤ Ad2(F )

S(F )r(v0)

= Ad2(F )

S(F )Φ(|y0 − x0|, 2|z0 − x0|) ≤ A

d(F )

S(F )φ(d(F )) ≤ A

d(F )

S(F ).

Therefore,

|tx0z0| ≤ |tx0z0

− tv0| + |tv0

| ≤ Ad(F )

S(F ).

Keeping in mind that d(F ) ≥ AS(F ) we obtain for (x, y) ∈ F the required estimate

|txy| ≤ |txy − tx0z0| + |tx0z0

|

≤ AΦ(|y − x|, 2|z0 − x0|) +Ad(F )

S(F )≤ A

d(F )

S(F )Φ(|y − x|, 3).

Thus, the vector field tu, u ∈ F , satisfies conditions (ii)(a)–(ii)(d) of Proposition

5.2.10 with a constant specified in (5.26). Applying the sufficiency part of Proposition

5.2.10 we finalize the proof of Proposition 5.2.11.

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