fractional order sobolev spacesweb.cs.elte.hu/blobs/diplomamunkak/msc_mat/2012/...fractional order...

Fractional order Sobolev spaces

Thesis

Matematikus MSc

Készítette: Gerencsér Máté

Témavezet®: Izsák Ferencadjunktus

Alkalmazott Analízis és Számításmatematikai Tanszék

Eötvös Loránd Tudományegyetem

Természettudományi Kar

Budapest, 2012

CONTENTS 1

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3. Classical scales of function spaces . . . . . . . . . . . . . . . . . . . . 5

3.1. Real interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2. Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3. Triebel-Lizorkin spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4. The fractionalization of Hcurl . . . . . . . . . . . . . . . . . . . . . . . 13

4.1. The scale Hscurl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2. The scale H .scurl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3. Fractionalization of the curl operator . . . . . . . . . . . . . . . . . . 19

4.4. Non-positive indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 1. INTRODUCTION

1. Introduction

The term fractional order Sobolev space might sound like a precise mathematical

concept but in fact it is not. There are several methods to ll in the gaps between

the traditional Sobolev spaces of integer order and in some cases the function spaces

obtained are equivalent, while in other cases they are not. Dierent approaches

focus on generalizing dierent properties of the Sobolev spaces and each has its own

advantage. These generalizations can be interesting and useful both theoretically and

in the applications as well. The aim of the thesis is to give an overview of these ideas

and apply these techniques to the a non-classical Sobolev space Hcurl.

The thesis is structured as follows. Chapter 2 summarizes the usual notions used

in the following. Chapter 3 describes the dierent scales of function spaces that

are usually referred to as fractional order Sobolev spaces, based on [1], [2], [5].

Chapter 4 examines the space of L2 functions whose curl is also in L2, and some

fractionalization problems regarding this space. The classical results of the topic

follows [4], the rest of the chapter is partially based on [3], partially my own work.

I would like to thank my supervisor, Izsák Ferenc, for his help throughout the

making of this thesis.

3

2. Preliminaries

We will use the following notions throughout the thesis.

A domain is an open subset Ω ⊂ Rn. A bounded domain is called Lipschitz, if

for every point x of its boundary ∂Ω there exists a neighborhood U of x such that

U ∩ ∂Ω is a graph of a Lipschitz-continous function.

For a normed space X by default ||·||X denotes its norm. The relation f(u) ≤Cg(u) (cf(u) ≤ g(u)) denotes that there exists a xed C > 0 (c > 0) for which the

inequality holds for all u from a given space that is always clear from the context.

This constant can change from line to line but is always independent from u. The

relation f(u) ∼ g(u) denotes that cg(u) ≤ f(u) ≤ Cg(u). Two norms ||·|| and ||·||′

on a normed space U are said to be equivalent if ||u|| ∼ ||u||′.The space D(Ω) consists of the compactly supoorted innitely many times dier-

entiable functions with the topology dened by the convergence: φn → φ if and only

if there is a compact set K ⊂ Ω with supp(φn), supp(φ) ⊂ K and for all multiindex α

∂αφn → ∂αφ uniformly. Its dual, D′ is the space of the continuous linear functionals,or, the distributions, with the weak topology. For every locally integrable f corre-

sponds a distribution with the eect φ→∫ ∑n

j=1 fjφj. For a normed function space

V in which D is imbedded, we dene its dual to be the subset of distributions which

extends uniquely to V .

The Schwartz space of functions u on Rn with sup |∂αu(x)xβ| < ∞ for all α, β

multiindices is denoted by S. The elements of S ′, the dual of S are called tempered

distributions. The Fourier transformation operator, which can be dened on S ′, isdenoted by F , and the notation u = F(u) is also used. On S, we dene F(φ)(ξ) =

(2π)−n/2∫

exp(i < x, ξ >)dx and we extend it to S ′ by Fu(φ) = u(Fφ).

Given a domain Ω ⊂ Rn we use the notation Lp(Ω) for the space of functions u

with ||u||pLp(Ω) =∫

Ω|u|p < ∞ (1 ≤ p < ∞). When it does not cause confusion, we

use the abbreviation ||·||Lp(Ω) = ||·||p. The standard Sobolev space of the functions

with their α-th partial derivatives in Lp(Ω) for all |α| ≤ k is denoted with W k,p(Ω),

accompanied with the norm ||u||Wk,p(Ω) =∑|α|≤k ||∂αu||p (k ∈ N, 1 ≤ p < ∞). Here

we use the derivatives in the distributional sense, i.e. for u ∈ D′, φ ∈ D, we dene

∂alphau(φ) = (−1)|α|u(∂α(φ)). We distinguish the special case Hk(Ω) = W k,2(Ω). We

also use the abbreviation Lp = Lp(Rn), and similarly with the Sobolev spaces. We

4 2. PRELIMINARIES

also use this convention for the spaces introduced later.

Given two Banach spaces X and Y their direct sum is denoted by X ⊕ Y and

consists of the formal sums x + y where x ∈ X, y ∈ Y , and ||x+ y||X⊕Y = (||x||2X +

||y||2Y )1/2. Note that changing 2 to any 1 ≤ p < ∞ we obtain equivalent Banach

spaces. If X and Y are Hilbert spaces, the norm can also be characterized by the

identities ||x+ 0||X⊕Y = ||x||X , ||0 + y||X⊕Y = ||y||Y , and the orthogonality of the

components. Given a closed subspace U in a Hilbert space PU denotes the projection

operator to U .

5

3. Classical scales of function spaces

This section aims to cover most of the possible denitions of fractional order

Sobolev spaces that can be found in the literature and describe their relations to

each other. To avoid confusion, we will omit the term fractional order Sobolev

space and use other common names for these spaces instead. We will formulate the

dierent but equivalent denitions in forms of theorems. For the detailed proofs we

refer to [1], [2], [5].

3.1. Real interpolation

Given two Banach spaces X0 and X1, both continously imbedded in a Banach

space X with a X0 ∩ X1 6= 0 - such a pair is called an interpolation couple -,

interpolation methods provide ways to construct intermediate spaces between them.

In many cases, including the ones we will deal with, X0 is continuously imbedded in

X1. The two main dierent methods are the real and complex interpolation but we

will now only go into details with the real method.

The intersection X0 ∩X1 and the algebraic sum X0 +X1 are themselves Banach

spaces with the norms

||u||X0∩X1= max||u||X0

, ||u||X1,

||u||X0+X1= inf||x0||X0

+ ||x1||X1|u = x0 + x1, x0 ∈ X0, x1 ∈ X1.

The intersection is continuously imbedded in Xj and Xj is continuously imbedded

in the algebraic sum for j = 0, 1. In general, we say that a Banach space U is

intermediate between X0 and X1 if X0 ∩ X1 is continuously imbedded in U and U

is continuously imbedded in X0 + X1. When X0 ⊂ X1, this equals the intuitive

requirement that and intermediate space has to be larger than X0 and smaller

than X1.

For any given x ∈ X0 +X1 ⊂ X, t ≥ 0, 1 ≤ p <∞ dene

Kp(t, x) = inf||x0||pX0+ ||tx1||pX1

|x = x0 + x1, x0 ∈ X0, x1 ∈ X11/p.

The usual approach takes p = 1, but it will be clear that all choice of p result in

6 3. CLASSICAL SCALES OF FUNCTION SPACES

equivalent Banach spaces. Since in the case of Hilbert spaces the choice p = 2 will

turn out to be the suitable one, we introduce the abbreviation K = K2. Now take

1 ≤ q <∞ and 0 < θ < 1. The interpolation space (X0, X1)θ,q consists of the vectors

x ∈ X0 +X1 for which

||x||q(X0,X1)θ,q=

∫ ∞0

[t−θK(t, x)]qdt

t<∞.

Theorem 3.1 Let (X0, X1) be an interpolation couple, 0 < θ < 1, and 1 ≤ q < ∞.

Then (X0, X1)θ,q is a Banach space and its norm satises

1√2||u||X0+X1

≤||u||(X0,X1)θ,q∫∞

0[t−θ min1, t]q dt

t

≤√

2 ||u||X0∩X1,

therefore (X0, X1)θ,q is an intermediate space between X0 and X1.

The extremal cases for q =∞ and/or θ = 0, 1 can also be dened, the correspond-

ing theorems are often trivial, but to avoid technical diculties we will not deal with

these cases.

Chopping the dening integral to integrals between 2j and 2j+1 one can show the

following discretization theorem.


Then for all x ∈ X0 +X1

||x||q(X0,X1)θ,q∼

∞∑j=−∞

2−jqθ(K(2j, x))q.


Then X0 ∩X1 is dense in (X0, X1)θ,q.

Let us introduce a functional similar to K, this time on X0 ∩X1:

J(t, u) = max||u||X0, ||tu||X1

.

This functional can be used to dene interpolation in another way resulting in equiv-

alent spaces, but more importantly, it can be used to formulate the Reiteration

3.1. Real interpolation 7

Theorem. First, we dene an intermediate space X to be in the class H(θ,X0, X1),

if

K(t, u) ≤ Ctθ ||u||X for all u ∈ X, and

||u||X ≤ Ct−θJ(t, u) for all u ∈ X0 ∩X1.

Lemma 3.4 Let (X0, X1) be an interpolation couple, 0 < θ < 1, and 1 ≤ q < ∞.

Then (X0, X1)θ,q ∈ H(θ,X0, X1).

Theorem 3.5 (Reiteration Theorem) Let (X0, X1) be an interpolation couple,

0 < λ < 1, 1 ≤ q <∞, 0 ≤ θ0 < θ1 ≤ 1, and Y0, Y1 intermediate spaces between X0

and X1 such that Yj ∈ H(θj, X0, X1), j = 0, 1. Let θ = (1− λ)θ0 + λθ1.Then

(Y0, Y1)λ,q = (X0, X1)θ,q.

The immediate consequence - and the reason for the name of the theorem - is

that with the previous notations

((X0, X1)θ0,q0 , (X0, X1)θ1,q1)λ,q = (X0, X1)θ,q

where 1 ≤ q0, q1 <∞ are arbitrary.

It is also an important property of the interpolation that its eect on the dual

spaces can be expressed easily.

Theorem 3.6 (Duality Theorem) Let (X0, X1) be an interpolation couple, 0 <

θ < 1, and 1 < q <∞ and assume that X0 ∩X1 is dense in both X0 and X1. Dene

q′ by 1/q + 1/q′ = 1. Then (X∗1 , X∗0 ) is also an interpolation couple and

(X∗1 , X∗0 )θ,q = (X0, X1)∗1−θ,q′ .

It is worth to note another simple identity in which this kind of change of param-

eters appear:

(X0, X1)θ,q = (X1, X0)1−θ,q.

When interpolating between two Hilbert spaces, it is natural to expect that the

result is a Hilbert space as well. By checking the paralelogram identity the following

theorem provides a sucient condition.

Theorem 3.7 Let (X0, X1) be an interpolation couple consisting of two Hilbert

spaces and 0 < θ < 1. Then (X0, X1)θ,2 is also a Hilbert space.

3.2. Besov spaces

The scale of Besov spaces is obtained by using the real interpolation method to

create intermediate spaces between Sobolev spaces. We have to note that in general

the classical Sobolev spaces are not closed under interpolation, i.e. W k,p(Ω) usually

does not equal to (Lp(Ω),Wm,p(Ω))k/m,q.

Lemma 3.8 Let Ω be a Lipschitz domain. Let 0 < k < m be integers and 1 ≤ p <∞.

Then

W k,p(Ω) ∈ H(k/m,Lp(Ω),Wm,p(Ω)).

Definition 3.9 Let 0 < s <∞, 1 ≤ p ≤ ∞, and 1 ≤ q <∞. Let m be the smallest

integer larger than s. Then the Besov space Bs,p,q(Ω) is dened by

Bs,p,q(Ω) = (Lp(Ω),Wm,p(Ω))s/m,q.

The Reiteration Theorem and the previous Lemma show us that in fact

Bs,p,q(Ω) = (W k,p(Ω),Wm,p(Ω))θ,q

for any k < s < m with k,m integers, 0 < θ < 1, and s = (1− θ)k + θm and also

Bs,p,q(Ω) = (Bs1,p,q1(Ω), Bs2,p,q2(Ω))θ,q

for any 0 < s1 < s < s2, 0 < θ < 1, and 1 ≤ q1, q2 ≤ ∞ with s = (1− θ)s1 + θs2.

The special cases Bs,p,p(Ω) are often referred as the Slobodeckij spaces and have

an important role of characterizing the traces of functions inWm,p(Ω). The trace of a

smooth function on Rn+1 is dened by restricting it to the subspace (x1, . . . , xn, 0),and the trace operator is extended in a usual way.

Theorem 3.10 Let 1 0 be an integer. Then u ∈ Bm−1/p,p,p(Rn)

if and only if u is the trace of a function in Wm,p(Rn+1).

3.2. Besov spaces 9

The Theorem does not apply in the case of p = 1. However, it is known that the

trace of a function from Wm,1 is in Wm−1,1. There is also a trace imbedding theorem

for Besov spaces.

Theorem 3.11 Let 1 ≤ p <∞, 1 ≤ q <∞, and s > 0 such that s− 1/p > 0. Then

the trace operator is continuous from Bs,p,q(Rn) to Bs−1/p,p,q(Rn−1).

Repeating taking traces gives imbedding theorems to spaces on Rk for suciently

large k. These theorems extend to traces on suciently smooth surfaces of suciently

high dimension as well. In case there exists a suitable extension operator from Ω,

these theorems also extend to functions in Bs,p,q(Ω).

It is sometimes useful to examine what more well-known spaces includes the Besov

space in question. This motivates the imbedding theorems similar to the following

one.

Theorem 3.12 Let 1 0, such that sp > n. Then Bs,p,q is

imbedded in the space of continuous and bounded functions.

The norms of the Besov spaces on Rn have a more intristic equivalent expressed

with the Lp-modulus of continuity. First, for a point h ∈ Rn and a function u ∈ L(R3)

dene uh to be the mapping x to u(x − h). Let ∆hu = u − uh, ωp(u, h) = ||∆hu||p,and for positive integers m let ω

(m)p (u, h) = ||(∆h)

mu||p.

Theorem 3.13 Let 1 s > 0 with m being an integer and

u ∈ Lp(Rn). Then u ∈ Bs,p,q(Rn) if and only if∫Rn

[|h|−sω(m)

p (u, h)]q dh|h|n

<∞.

Moreover, the q-th root of the expression above is equivalent to ||·||Bs,p,q .

For non-integers s = bsc+ s, where bsc ∈ N and s ∈ (0, 1), a closely related

intristic norm can be dened using the bsc-th order derivatives:

Theorem 3.14 Let 1 0, and u ∈ W bsc,p(Rn). Then

u ∈ Bs,p,q(Rn) if and only if

∑|α|=bsc

∫Rn

(∫Rn|∂αu(x)− ∂αu(x− h)|p

)q/p|h|−sq dh

|h|n<∞.

There is another equivalent intristic norm that will show some kind of relation

between the Besov spaces and the Triebel-Lizorkin spaces dened in the next subsec-

tion. To this end, rst let Φ be an even function on the real line with the properties

Φ(t) = 1 for |t| ≤ 1, Φ(t) = 0 for |t| ≥ 2, and |Φ(t)| ≤ 1 for all t. For each integer i

let φi(t) = Φ(t/2i+1)−Φ(t/2i) and for any ξ ∈ Rn let ψi(ξ) = φi(|ξ|). Finally, denethe operator Tiu = F−1(ξ → ψi(ξ)u(ξ)). One can regard the functions Tiu as dyadic

parts of u with nearly disjoint frequencies.

Theorem 3.15 The norm ||u||Bs,p,q(Rn) is equivalent to[∞∑

j=−∞

(∫Rn

(1 + 2sj)p|Tju(x)|pdx)q/p]1/q

.

3.3. Triebel-Lizorkin spaces

Definition 3.16 Let 0 < s < ∞, 1 ≤ p < ∞, and 1 ≤ q < ∞. Then the Triebel-

Lizorkin space F s,p,q(Ω) is dened by

F s,p,q(Rn) = u : ||u||F s,p,q(Rn) =

∫Rn

(∞∑

j=−∞

(1 + 2sj)q|Tju(x)|q)p/q

dx

1/q

<∞.

Clearly F s,p,p = Bs,p,p, and therefore the Slobodeckij spaces are included in this

scale as well. Another important special case is obtained by generalizing the well-

known fact that for any k integer u ∈ W k,p if and only if the function ξ → (1 +

|ξ|2)k/2u(ξ) is the Fourier transform of a function from Lp. The spaces we get by

changing k to any s ≥ 0 in this property are often referred to as the Bessel potential

spaces.

Theorem 3.17 Let 0 < s < ∞ and 1 ≤ p < ∞. Then u ∈ F s,p,2 if and only if the

function ξ → (1 + |ξ|2)s/2u(ξ) is the Fourier transform of a function from Lp.

Since (1+|ξ|2)s/2 can be bounded from above and below by constant times 1+|ξ|s,the previous condition can be rephrased that require u and ξ → F−1|ξ|su(ξ) to be

in Lp. The latter expression is also known as the fractional Laplacian of u, and so

the Bessel potential spaces can be obtained by generalizing the denition of Sobolev

3.3. Triebel-Lizorkin spaces 11

spaces by the derivatives of the function. The fractional Laplacian is one of the

possible generalizations of the dierentiation operator (see [7], Chapter 2), it simply

fractionalizes the positive operator −∆. It also follows from the previous Theorem

that Fm,p,2 = Wm,p for integers m.

It also turns out that the Bessel potential spaces can be obtained from the classical

Sobolev spaces the same way as the Besov spaces if we use the complex interpolation

method instead of the real interpolation.

Definition 3.18 Let (X0, X1) be an interpolation couple. Let A denote the collec-

tion of bounded analytic functions f from the strip θ + iτ |0 < θ < 1 to X0 +

X1 that extend continously to the boundary with the property f(j + τ) ∈ Xj and

||f(j + iτ)||Xj → 0 as |τ | → ∞, for j = 0, 1. For 0 < θ < 1 the complex interpola-

tion space between X0 and X1 is

[X0, X1]θ = u ∈ X0 +X1|∃f ∈ A : f(θ) = u

with the norm

||u||[X0,X1]θ= infmaxsup

τ||f(iτ)||X0

, supτ||f(1 + iτ)||X1

|f(θ) = u.

Although we omit the detailed description of this method, it is worth to note

that it has similar properties to the real method. For example the analogues of the

Reiteration Theorem and the Duality Theorem hold. We also have the following

identities that show the connection between the two methods

(X0, X1)θ,1 ⊂ [X0, X1]θ ⊂ (X0, X1)θ,∞,

([X0, X1]θ0 , [X0, X1]θ1)λ,q = (X0, X1)(1−λ)θ0+λθ1,q,

[(X0, X1)θ0,q0 , (X0, X1)θ1,q1 ]λ = (X0, X1)(1−λ)θ0+λθ1,q,

where q is dened by 1/q = (1− λ)/q0 + λ/q1.

Theorem 3.19 Let 0 < s <∞, 0 < θ < 1, m ≤ k integers with (1− θ)m+ θk = s,

and 1 ≤ p <∞. Then

F s,p,2 = [Wm,p,W k,p]θ.

Similar imbedding theorems holds for the Triebel-Lizorkin spaces as for the Besov

spaces. For example, analogously to the one mentioned in the previous subsection,

the following theorem holds.

Theorem 3.20 Let 1 0, such that sp > n. Then F s,p,q is

imbedded in the space of continuous and bounded functions.

Finally, we summarize the relations between the spaces introduced:

F s,p,q ⊂ Bs,p,q if q ≤ p,

Bs,p,q ⊂ F s,p,q if p ≤ q,

Bs,p,q0 ⊂ Bs,p,q1 if q0 ≤ q1,

F s,p,q0 ⊂ F s,p,q1 if q0 ≤ q1.

In the special case p = q = 2, Bs,2,2 = F s,2,2 with the special cases Fm,2,2 = Hm in

case m is integer, and furthermore we have

Bs,2,2 = (Bs0,2,2, Bs1,2,2)θ,2 = [Bs0,2,2, Bs1,2,2]θ

for 0 < s0 < s < s1 <∞ with (1− θ)s0 + θs1 = s.

Remark. Some of the denitions and the theorems were stated only in the case

Ω = Rn. These properties can be extended to more general domains via the use of an

extension operator. In [6] it is shown that there exists an extension operator E such

that it simultaneously and boundedly extends functions in Bs,p,q(Ω) to Bs,p,q(Rn)

and F s,p,q(Ω) to F s,p,q(Rn) if the domain Ω is nice enough, for example, if it is a

Lipschitz domain. Here F s,p,q(Ω) is dened as restrictions of functions on F s,p,q(Rn).

The spaces Bs,p,q(Ω) were already dened, but from the existence of this extension

operator it follows that they can also be dened through restriction.

13

4. The fractionalization of Hcurl

In this section we deal with the possibilities of the fractionalization of the non-

standard Sobolev space Hcurl = u ∈ (L2)3|curl(u) ∈ (L2)3. This space plays an

important role in the theory of Maxwell equations. Even the extension to integer

orders is not evident, we present two approaches.

First recall some properties of the curl operator. It is dened by

curl(u) =

(∂u3

∂2

− ∂u2

∂3

,∂u1

∂3

− ∂u3

∂1

,∂u2

∂1

− ∂u1

∂2

)where derivatives are understood in the distributional sense. Its Fourier counterpart

is the vectorial product with the longitudinal direction:

F(curl(u))(ξ) = i(ξ × u(ξ)).

The same formula holds if we replace curl by (curl)j, ξ× by ξ×j, and i by ij for anyj ∈ N. Here the operator ξ×j is dened by

ξ ×j f(ξ) = ξ × · · · ξ×︸︷︷︸j

f(ξ)

and similarly for (curl)j. The space Hcurl can be decomposed

Hcurl = Ker(curl)⊕Ker(curl)⊥

where the second component is compactly imbedded in (L2)3. This is known as the

Helmhlotz decomposition. The Fourier transform of a function in the kernel of the

curl points in the longitudinal direction almost everywhere. Conversely, if a function

from (L2)3 has a Fourier transform that points in the longitudinal direction, its curl

is zero. Therefore it is easy to see that if u ∈ Ker(curl)⊥, then its Fourier transform

has to point perpendicular to the longitudinal direction.

It is known that the sequence called the De Rahm-diagram

(H1)3 grad−→ Hcurlcurl−→ Hdiv

div−→ (L2)3 (1)

14 4. THE FRACTIONALIZATION OF HCURL

has the property that the image of each operator is included in the kernel of the next

one, where

Hdiv = u ∈ (L2)3|div(u) =∂u1

∂1

+∂u2

∂2

+∂u3

∂3

∈ L2.

Similarly, if our initial space is (L2(Ω))3, we can dene Hcurl(Ω), Hdiv(Ω) for any

domain. Assume that Ω is bounded and has the property that there exist L open

connected surfaces Σ1,Σ2, . . .ΣL such that

• Σl is an open part of a smooth surface,

• ∂Σl ⊂ ∂Ω,

• Σl ∩ Σm = ∅ if l 6= m,

• For any point x ∈ ∂Ω there is an integer rx ∈ 1, 2 and a ρx > 0 such that for

all 0 < ρ < ρx the intersection of Ω with the ball with center x and radius ρ

has rx connected components, each one being a Lipschitz domain.

Under these conditions the diagram (1) also has the property that the image of each

operator is a closed subspace of nite codimension in the kernel of the next operator

[4].

4.1. The scale Hscurl

If we consider the curl operator as a kind of dierentiation, it is natural to intro-

duce the following spaces

Hncurl = u ∈ (L2)3|curlj(u) ∈ (L2)3 for j = 1, 2, . . . n.

Hncurl is a Hilbert space with the norm ||u||2Hn

curl=∑n

j=0 ||curlj(u)||22. We can extend

this denition for any non-integer s > 0 by dening

Hscurl = ((L2)3, H

dsecurl)s/dse,2 (2)

where dse is the smallest integer larger than s.

Before moving on to examine these spaces further, we establish some simple prop-

erties of the real interpolation.

4.1. The scale Hscurl 15

Proposition 4.1 Let X, Y, V,W be Banach spaces, 1 ≤ q < ∞, and 0 < θ < 1.

Then

(X ⊕ V, Y ⊕W )θ,q = (X, Y )θ,q ⊕ (V,W )θ,q

with equivalent norms. In case q = 2, the same holds for Hilbert spaces.

Proof. The equality of the sums simply follows from considering

1√2

(t ||x||X + ||y||Y + t ||v||V + ||w||W ) ≤ t ||x+ v||X⊕V + ||y + w||Y⊕W ≤

≤ t ||x||X + ||y||Y + t ||v||V + ||w||W .

After taking inmums, multiplying by tθ, taking q-th powers, and integrating with

respect to the measure dtton [0,∞), we get the inequalities required.

In case q = 2 it remains to show that for any z + u ∈ (X ⊕ V, Y ⊕W )θ,q, z is

perpendicular to u.

4 < z, u >= ||z + u||2(X⊕V,Y⊕W )θ,q− ||z − u||2(X⊕V,Y⊕W )θ,q

=

=

∫ ∞0

t−2θ((K(t, z + u))2 − (K(t, z − u))2

) dtt.

Note that the inmum dening K can be decomposed:

K(t, z + u) = inf||t(x+ v)||2X⊕V + ||y + w||2Y⊕W |z + u = (x+ v) + (y + w)1/2 =

= inf||tx||2X + ||y||2Y |z = x+ y1/2 + inf||tv||2V + ||w||2W |u = v + w.

K(t, z − u) has exactly same form, except that in the second inmum v and w are

multiplied by −1. This does not aect the value of the inmum, therefore K(t, z +

u))− (K(t, z − u)) = 0 and < z, u >= 0.

Proposition 4.2 Let U, V,W be subspaces of a Hilbert space X, and suppose that

W is closed, 1 ≤ q <∞, and 0 < θ < 1. Then

(U, V )θ,q ∩W = (U ∩W,V ∩W )θ,q

with equivalent norms.

Proof. Set w ∈ W . For every decomposition w = u+ v we can take w = (PU∩Wu+

PV ∩Wv) + (PU∩W⊥u+PV ∩W⊥v) instead. The second term is in W⊥ by defnition, and

it is also a dierence of two elements of W , therefore it is in W ∩W⊥ and thus it is

zero. The norm of the components can only decrease with this modication, therefore

KU∩W,V ∩W (t, w) ≤ KU,V (t, w), while the other inequality is trivial. The equivalence

of the two K functionals then implies the equivalence of the norms.

Proposition 4.3 Let U, V be Banach spaces with U compactly imbedded in V , 1 ≤q <∞, and 0 < θ < 1. Then (U, V )θ,q is also compactly imbedded in V .

Proof. Let zn be a sequence in the unit ball of (U, V )θ,q. Using the discrete

version of interpolation and K1 there are unj ∈ U and vnj ∈ V such that unj + vnj = zn

and∞∑

j=−∞

[2−jθ(2j

∣∣∣∣unj ∣∣∣∣U +∣∣∣∣vnj ∣∣∣∣V )

]q≤ 2 ||zn||(U,V )θ,q

≤ 2.

We have∣∣∣∣unj ∣∣∣∣U ≤ 2j(θ−1)+1, therefore unj ⊂ V has a Cauchy subsequence for all j

and with the diagonal argument we get a subsequence zmk such that umkj ⊂ V

is Cauchy for all j. Fix ε > 0 and choose j so that 2jθ < ε. Then we have∣∣∣∣vnj ∣∣∣∣V <

2ε and therefore if N is chosen so that for mk,ml > N∣∣∣∣umkj − umlj ∣∣∣∣V < ε, then

||zmk − zml ||V < 5ε.

This statement is true in a more general way, but this weaker version easily

implies a stronger one. Note that from the Schauder theorem it follows that if U is

compactly imbedded in V then V ∗ is compactly imbedded in U∗. In case U and V

are reexive spaces we can apply the previous proposition to the dual spaces. By

the Duality Theorem we obtain that U is compactly imbedded in (U, V )θ,q and by

the Reiteration Theorem we can conclude that (U, V )θ1,q is compactly imbedded in

(U, V )θ2,q for 0 < θ1 < θ2 < 1.

Let Hn0,curl denote the orthogonal complement of Ker(curl) in Hn

curl. Note that

the inner product, and therefore the orthogonality, is the same in every Hncurl if one

of the functions is in Ker(curl). Proposition 4.1 shows that it suces to interpolate

between these spaces. It turns out that they are subspaces of the traditional Sobolev

spaces.

4.2. The scale H .scurl 17

Proposition 4.4

(Hn)3 ∩Ker(curl)⊥ = Hn0,curl


Proof. Let u ∈ Ker(curl)⊥. For the Fourier transform of such a function ξ⊥u(ξ)

holds almost everywhere, therefore

||u||Hn0,curl

∼n∑j=0

∣∣∣∣curlj(u)∣∣∣∣

2=

n∑j=0

∣∣∣∣ξ → ξ ×j u(ξ)∣∣∣∣

2=

=n∑j=0

∣∣∣∣ξ → |ξ ×j u(ξ)|∣∣∣∣

2=

n∑j=0

∣∣∣∣ξ → |ξ|j|u(ξ)|∣∣∣∣

2∼∣∣∣∣ξ → (1 + |ξ|2)n/2|u(ξ)|

∣∣∣∣2

=

=∣∣∣∣ξ → (1 + |ξ|2)n/2u(ξ)

∣∣∣∣2∼ ||u||(Hn)3 .

Using the results from the interpolation of Sobolev spaces we can write the fol-

lowing

Corollary 4.5

Hscurl = Ker(curl)⊕ ((Bs,2,2)3 ∩Ker(curl)⊥) (3)

as Hilbert spaces, where by Proposition 4.3 the second component is compactly imbed-

ded in (L2)3.

This is a slightly generalized form of the Helmholtz decomposition. The conse-

quence of this decomposition is that these spaces "behave well" under interpolation,

i.e.

(Hs1curl, H

s2curl)θ,2 = Hs

curl

for any 0 ≤ s1 < s < s2 with (1− θ)s1 + θs2 = s.

4.2. The scale H .scurl

There is another common way to extend the denition of Hcurl to integer indices,

found in e.g [4]. In this case we are not interested in the higher order curls of the

functions, but instead the smoothness of the function and the curl of the function is

simultaneously prescribed. Dene H .ncurl by

H .ncurl = u ∈ (Hn)3|curl(u) ∈ (Hn)3.

These are also Hilbert spaces with the norm ||u||2H.ncurl

= ||u||2(Hn)3 +||curl(u)||2(Hn)3 .

It is possible to derive a similar decomposition to (3) for these spaces. Obviously, if

u ∈ Ker(curl) then u ∈ H .ncurl if and only if u ∈ (Hn)3.

Proposition 4.6 Let Ker(curl)⊥ denote the orthogonal complement of Ker(curl) in

(L2)3. Then

H .ncurl ∩Ker(curl)⊥ = (Hn+1)3 ∩Ker(curl)⊥


Proof. Similarly to the proof of Proposition 4.4, consider a function u ∈ Ker(curl)⊥

for which therefore ξ⊥u(ξ) holds. Using again the equivalent Sobolev norm and the

unitarity of the Fourier transform we can write

||u||H.ncurl∼∣∣∣∣ξ → (1 + |ξ|2)n/2|u(ξ)|

∣∣∣∣2

+∣∣∣∣ξ → (1 + |ξ|2)n/2|ξ × u(ξ)|

∣∣∣∣2

=

=∣∣∣∣ξ → (1 + |ξ|2)n/2|u(ξ)|

∣∣∣∣2

+∣∣∣∣ξ → |ξ|(1 + |ξ|2)n/2|u(ξ)|

∣∣∣∣2∼

∼∣∣∣∣ξ → (1 + |ξ|2)(n+1)/2|u(ξ)|

∣∣∣∣2∼ ||u||(Hn+1)3 .

Observe that if u ∈ Ker(curl)∩ (Hn)3 and v ∈ Ker(curl)⊥ ∩ (Hn)3, then they are

orthogonal with respect to the (Hn)3 inner product as well. Indeed, taking partial

derivatives from the functions does not change the direction of the Fourier transform

of the function, and thus F(∂αu) points in the longitudinal direction, and F(∂αv)

points perpendicular to the longitudinal direction. Therefore < F(∂αu),F(∂αv) >=

0 pointwise. We thus get the decomposition

H .ncurl = ((Hn)3 ∩Ker(curl))⊕ ((Hn+1)3 ∩Ker(curl)⊥)

4.3. Fractionalization of the curl operator 19

as Hilbert spaces and we can again conclude that

(H .mcurl, H

.ncurl)θ,2 = H .s

curl

for any 0 ≤ m < s < n integers with (1−θ)m+θn = s. We can also extend this scale

to any non-integer s by using the above interpolation equation as a denition. By the

same interpolation properties as we used in (3), we then obtain the decomposition,

now for all s ≥ 0,

H .scurl = ((Bs,2,2)3 ∩Ker(curl))⊕ ((Bs+1,2,2)3 ∩Ker(curl)⊥).

4.3. Fractionalization of the curl operator

Similarly to the fractional versions of the Laplacian operator, it is possible to

fractionalize the curl as well through fractionalizing the appropriate operator acting

on the Fourier transform. A natural expectation for these operators to form a semi-

group, and also we wish that for integer indices we get back the curln operator dened

earlier. Let u ∈ Ker(curl)⊥ ∩ S and examine how the curl operator changes u at a

point ξ. Both u(ξ) and ξ× u(ξ)/i falls in the subspace ξ⊥. In fact ξ× u(ξ)/i = Au(ξ),

where

A = Aξ = |ξ|Rπ/2 = |ξ|

(cos(π/2) − sin(π/2)

sin(π/2) cos(π/2)

)is a linear operator acting on ξ⊥. Similarly, ξ ×n u(ξ)/in = Anu(ξ). The s-th power

of A can be easily dened by As = |ξ|sRsπ/2 for any s ≥ 0 and this leads to the

denition

curls(u) = F−1(ξ → isAsξu(ξ)).

This denition can be extended to any u ∈ Ker(curl)⊥ ∩ (L2)3: since u is locally

integrable, so is ξ → isAsξu(ξ) and therefore we can take its inverse Fourier transform.

In fact this denition extends to any tempered distribution having locally integrable

Fourier-transform. It is easy to see that curls(curlt(u)) = curls+t(u) and we dened

curln so that coincides with the n-th power of the curl operator if n is an integer.

If u ∈ Ker(curl)∩ (L2)3 then we can simply take curls(u) = 0 for s > 0. Then we

can extend curls to (L2)3 linearly. It is clear that the properties curls(curlt(u)) =

curls+t(u) and curln = (curl)n still hold. This extension therefore satises our ex-

pectations and also, it is closely related to the scale of spaces Hscurl.

Proposition 4.7 Let u ∈ (L2)3. Then u ∈ Hscurl if and only if curls(u) ∈ (L2)3.

This shows that this fractionalization of the curl could also be used to give another

equivalent denition of Hscurl quite analogously to the traditional Sobolev spaces.

Proof. Clearly it is enough to show the claim for u0 = PKer(curl)⊥u. By the de-

composition (3), u0 ∈ (Bs,2,2)3 = (F s,2,2)3, and we can use the equivalent norm of

Proposition 3.17. Therefore,

||u0||(L2)3 + ||u0||Hscurl∼ ||ξ → u0(ξ)||2 +

∣∣∣∣ξ → (1 + |ξ|2)s/2u0(ξ)∣∣∣∣

2∼

∼ ||ξ → u0(ξ)||2 + ||ξ → ||ξ|su0(ξ)|||2 =

= ||ξ → u0(ξ)||2 +∣∣∣∣ξ → |is|ξ|sRsπ/2u0(ξ)|

∣∣∣∣2

= ||u0||2 + ||curls(u0)||2 .

Note that this calculation shows that, in fact, if we introduce the norm (||u||22 +

||curls(u)||22)1/2 on Hscurl, we obtain an equivalent Hilbert space.

The analogue of the diagram (1) can be written as

(H1)3 grad−→ Hscurl

curls−→ Hdivdiv−→ (L2)3

for any s > 0, and it preserves the property that the image of each operator is

included in the kernel of the next one. Indeed, the kernel of curls is the same for

any s > 0, namely it consists of the functions u ∈ (L2)3 whose Fourier transform

points in the longitudinal direction almost everywhere. On the other hand, the

kernel of div consists of the functions u ∈ (L2)3 whose Fourier transform points in

perpendicular to longitudinal direction. By denition, all functions of form curlsu

has this property. However, using the fractional curl it is easy to show that the

inclusion Im(curls) ⊂ Ker(div) is not of nite codimension.

Proposition 4.8 Set 0 < T ≤ ∞ and let the family of linear operators Css∈(0,T )

acting on the spaces Vss∈(0,T ) such that CsCt is dened and equals to Cs+t for all

choice of s, t > 0 with s + t < T . Suppose furthermore that Ker(Cs) = Ker(Ct) for

4.3. Fractionalization of the curl operator 21

all s, t > 0. Then Im(Ct) ⊂ Im(Cs) for all 0 < s < t < T and the inclusions are

either strict for every choice of s and t, or are trivial for every choice.

Proof. The rst part is trivial from Ctu = Cs(Ct−su). For the second part, it

is enough to check the actions of Cs on Vs/Ker(Cs), thus we can assume that the

operators are injectives. Suppose that the inclusion Im(Ct) ⊂ Im(Cs) is srtict for a

xed 0 < s < t. For any r > q > t x m such that (t − s)/m ≤ r − q. At least forone j ∈ 1, 2, . . . ,m one of the inclusions Im(Cs+j(t−s)/m) ⊂ Im(Cs+(j−1)(t−s)/m) is

strict, that is, there is a v such that v = Cs+(j−1)(t−s)/mu, but there is no such w that

v = Cs+j(t−s)/mw. Consequently there is no such w that v = Cs+(j−1)(t−s)/m+(r−q)w

either. Thus v′ = Cq−(s+(j−1)(t−s)/m)v equals to Cqu, but doesnt equal to Crw for any

w, and thus Im(Cr) ⊂ Im(Cq). We can conclude that there exists an s0 such that

the inclusion of images is trivial if both indices are smaller than s0, and strict if both

indices are larger than s0. On the other hand, if Im(Cs) = Im(C2s) = Im(CsCs) for

a xed s, then Cs maps Im(Cs) onto itself, therefore Im(Cms) = Im(Cns) for any

n,m integers. Hence s0 can be only 0 or T , which proves the statement.

It now suces to nd an u ∈ Im(curl)\Im(curl2), and then the Proposition yields

that all the intermediate inclusions in

Im(curls) ⊂ Im(curls/2) ⊂ · · · ⊂ Im(curls/2n

) ⊂ · · · ⊂ Ker(div)

are strict, therefore the codimension of Im(curls) in Ker(div) is innite. Consider

the function v = F−1(ξ → χ|ξ|<1f(ξ)/|ξ|) where |f(ξ)| = 1 and its direction is chosen

in a measurable way such that it points perpendicular to the longitudinal direction.

The function we are looking for is u = curl(v). Clearly u and v are in (L2)3, so

u ∈ Im(curl). However, u /∈ Im(curl2), or equivalently, v /∈ Im(curl). Suppose

v = curl(w), and consider w′ = PKer(curl)⊥w. We know that |ξ×w′(ξ)| = |ξ||w′(ξ)| =

χ|ξ|<1/|ξ|, and therefore |w′(ξ)| = χ|ξ|<1/|ξ|2, so w′ is not in (L2)3. Consequently,

neither is w′, or, by ||w||2 ≥ ||PKer(curl)⊥w||2, neither is w.

Our denition of the fractional curl, just like usually the fractional derivatives,

is not local. Therefore there is no guarantee that the fractional curl of a compactly

supported function will also have the same support. If we wish to fractionalize the

operator curl : Hcurl(Ω) → L2(Ω) in the case of for example a simply connected

bounded Lipschitz-domain, then by the properties mentioned at the beginning of the

chapter and the arguments above, the image of curls would have to be the same for

all s > 0. Therefore if a counterexample like in the preceeding is found with compact

support, then it proves that such fractionalization can not exist.

As an interesting application, it is described in [3] that with the fractionalization

of the curl it is possible to generalize the principle of duality in electromagnetics.

The source-free Maxwell equations in vacuum for the time-harmonic case with the

time dependence e−iωt can be written in the form

1

ik0

curl(η0H) = −E

1

ik0

curl(E) = η0H

div(η0H) = 0

div(E) = 0.

Using the semigroup-property and div(curls) ≡ 0 we get that if E andH are solutions,

then so are their fractional duals

Efd = c · curls(E), η0Hfd = c · curls(η0H)

for any s > 0 and c constant. If we choose c = 1/(ik0)s, then for the special case

s = 1 we obtain Efd = η0H and η0Hfd = −E, the dual elds of the original solution.

4.4. Non-positive indices

The method in the preceeding for the fractionalization of the vectorial product,

and therefore the curl in fact works for negative, or even complex s. Since in this

case ensuring the semigroup property requires curl0 = curlscurl−s, we have to dene

curl0 to be the projection to Ker(curl)⊥ instead of the more natural choice of the

identity operator.

By the virtue of Proposition 4.7, we could dene Hscurl to be the space of functions

u ∈ L2 such that curls ∈ L2. Observe that for any s, the eigenvalues of Rsπ/2 are

nonzeros and are same for all ξ, that is, |Rsπ/2x| ∼ |x|. Thus, if <(s1) = <(s2) = σ,

4.4. Non-positive indices 23

then

|is1As1ξ u(ξ)| = |ξs1 ||Rs1π/2u(ξ)| = |ξ|σ|Rs1π/2u(ξ)| ∼ |ξ|σ|Rs2π/2u(ξ)| = |is2As2ξ u(ξ)|.

It follows that Hs1curl = Hs2

curl = Hσcurl, so the introduction of non-real complex indices

does not yield new spaces. On the other hand, even for negative s, sinceHscurl ⊂ (L2)3,

this denition would provide a scale that clearly lacks the property (Hscurl)

′ = H−scurl.

Since the analogue property is held by the classical Sobolev spaces, and even the

versions of Besov-, and Triebel-Lizorkin spaces for negative indices, we will provide a

new denition. First let us determine these dual space similarly to the classical case

([1], Theorem 3.9).

Proposition 4.9 Set s > 0. The dual space of Hscurl is given by

u ∈ S ′|u = u1 + curlsu2|u1, u2 ∈ (L2)3

.

Proof. Let us dene the operator curls∗ in exactly the same way as curls, with

the only exception being using the transpose of the matrix Rsπ/2. For now the only

interesting property is that

||curls∗(u)||2 = ||curls(u)||2 .

This follows from that since Rsπ/2 is a rotation, |Rsπ/2ξ|2 = |RTsπ/2ξ|2 = |ξ|2.

Now consider the space V = Q1∪Q2 consisting of two distinct copies of Rn and let

P be the mapping from Hscurl to L2(V ) such that Pu|Q1 = u and Pu|Q2 = curls∗(u).

By the equivalent norm in Proposition 4.7, P is an isometric isomorphism between

its domain and its range W . For any L a linear functional on Hscurl we can dene L∗

on W by L∗(Pu) = Lu. By the Hahn-Banach theorem L∗ can be extended in a norm

preserving way to a linear functional on L2(V ). Denoting this extension by L′, we

can use the Riesz Representation theorem for L′. That is, there exists a v ∈ L2(V )

such that L′z =< z, v > for all z ∈ L2(V ), in particular

Lu = L′(Pu) =< u, v1 > + < curls∗(u), v2 > (4)

for all u ∈ Hscurl where the latter inner products are understood in the (L2(Rn))3 sense,

and vj = v|Qj for j = 1, 2. Let us use the notation Tf for the regular distribution for

any given locally integrable function f . That is, Tf (φ) =∫ ∑3

j=1 fjφj. Consider the

distribution

T = Tv1 + curls(Tv2). (5)

We now show that T is extended to Hscurl by L. Taking φ ∈ S, we clearly have

Tv1(φ) =< φ, v1 >, and

curls(Tv2)(φ) = F(curls(Tv2)(F−1(φ)) =

∫is|ξ|s

3∑j=1

(Rsπ/2v2)j(ξ)φj(−ξ) =

=

∫is|ξ|sv2(ξ)TRT

sπ/2φ(−ξ) =

∫is|ξ|s

3∑j=1

(v2)j(ξ)(RTsπ/2φ)j(−ξ) =

= F(v2)(F−1(curls∗(φ)) =< curls∗(φ), v2 > .

Comparing this with (4), we get T (φ) = L(φ), so L indeed extends T .

On the other hand, if T is of the form (5), we need to show that it has a unique

extension to Hscurl. Since S is dense in (L2)3, which is dense in Hs

curl, for any xed

u ∈ Hscurl we can nd a sequence φn∞n=1 ⊂ S converging to it. We can write

|T (φk)− T (φl)| ≤ | < φk − φl, v1 > |+ | < curls∗(φk − φl), v2 > | ≤

≤ ||φk − φl||2 ||v1||2 + ||curls(φk − φl)||2 ||v2||2 ≤ ||φk − φl||Hscurl||v||L2(V ) .

Thus T (φn) is a Cauchy sequence in C and so converges to a limit that we can

denote by L(u) since it is clear that we obtain the same limit to any other sequence

ψn ⊂ S converging to u. The functional L is linear and also bounded, since

|L(u)| = limn→∞

|T (φn)| ≤ limn→∞

||φn||Hscurl||v||L2(V ) = ||u||Hs

curl||v||L2(V ) .

We can now extend the scale Hscurl to negative indices in a natural way, while also

ensuring the expectations for duality. Let A denote the subspace of S ′ with distri-

butions whose Fourier transform is locally integrable and its longitudinal component

is in (L2)3. For any s ∈ R set As = A ∩ u ∈ S ′|curls(u) ∈ (L2)3.

4.4. Non-positive indices 25

Proposition 4.10 The scale

Hscurl =

(L2)3 ∩ As if s ≥ 0

(L2)3 + As if s ≤ 0

coincides with the initial denition (2) for s ≥ 0 and has the property (Hscurl)

′ = H−scurl.

Proof. The rst claim is simply a rephrasing of Proposition 4.7. The only dierence

is the intersection with A which does not change our space since Hscurl ⊂ (L2)3 ⊂ A

for s ≥ 0. The second part is Proposition 4.9 for s ≥ 0, while the case s ≤ 0 follows

from the reexivity of Hilbert spaces.

26 REFERENCES

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Berlin-New York (1976)

[3] N. Engheta, Fractional curl operator in electromagnetics, Microw. Opt. Technol.

Lett., 17: 86-91 (1998)

[4] P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press,

Oxford (2003)

[5] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators,

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