weighted local morrey spacesexample, local morrey spaces play important role to describe the real...

Annales Academiæ Scientiarum FennicæMathematicaVolumen 45, 2020, 67–93

WEIGHTED LOCAL MORREY SPACES

Shohei Nakamura, Yoshihiro Sawano and Hitoshi Tanaka

Tokyo Metropolitan University, Department of Mathematics and Information SciencesMinami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan; [email protected]

Tokyo Metropolitan University, Department of Mathematics and Information SciencesMinami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan; [email protected]

and Department of Mathematical Analysis and Theory of FunctionsPeoples Friendship University of Russia (RUDN University)

6 Miklukho-Maklay St, Moscow, 117198 Russia

National University Corporation Tsukuba University of TechnologyResearch and Support Center on Higher Education for the Hearing and Visually Impaired

Kasuga 4-12-7, Tsukuba 305-8521, Japan; [email protected]

Abstract. We discuss the boundedness of linear and sublinear operators in two types of

weighted local Morrey spaces. One is defined by Natasha Samko in 2008. The other is defined by

Yasuo Komori-Furuya and Satoru Shirai in 2009. We characterize the class of weights for which

the Hardy–Littlewood maximal operator is bounded. Under a certain integral condition it turns

out that the singular integral operators are bounded if and only if the Hardy–Littlewood maximal

operator is bounded. As an application of the characterization, the power weight function | · |α is

considered. The condition on α for which the Hardy–Littlewood maximal operator is bounded can

be described completely.

1. Introduction

The aim of this paper is to characterize the class of weights for which the Hardy–Littlewood maximal operator M is bounded on the weighted local Morrey spaceLMp

q(1, w) of Samko type and on the weighted local Morrey space LMpq(w,w) of

Komori–Shirai type. A similar characterization is obtained for the singular integraloperators, the fractional integral operators and the fractional maximal operators.Here and below by a weight we mean a locally integrable function on R

n which isalmost everywhere positive.

We shall consider all cubes in Rn which have their sides parallel to the coordinate

axes. We denote by Q the family of all such cubes. For a cube Q ∈ Q we use ℓ(Q) todenote the sides length of Q, c(Q) to denote the center of Q, |Q| to denote the volumeof Q and cQ to denote the cube with the same center as Q but with side-length cℓ(Q).

The class Ap with 1 < p < ∞ is defined to the set of all weights w for which

supQ∈Q

1

|Q|

ˆ

Q

w(x) dx

(ˆ

Q

w(x)−1

p−1 dx

)p−1

< ∞.

This class Ap, initiated by Muckenhoupt [42], characterizes the condition for whichthere exists a constant C > 0 such that

ˆ

Rn

Mf(x)pw(x) dx ≤ C

ˆ

Rn

|f(x)|pw(x) dx

https://doi.org/10.5186/aasfm.2020.45042010 Mathematics Subject Classification: Primary 42B25, 42B35, 26A33.Key words: Local Morrey spaces of Samko type, local Morrey spaces of Komori–Shirai type,

weights.

68 Shohei Nakamura, Yoshihiro Sawano and Hitoshi Tanaka

for all measurable functions f , where M is the Hardy–Littlewood maximal operatordefined by

Mf(x) ≡ supQ∈Q

χQ(x)

|Q|

ˆ

Q

|f(y)| dy, x ∈ Rn.

The Hardy–Littlewood maximal operator M plays a fundamental role in harmonicanalysis. The Riesz transform, which is given by

Rjf(x) ≡ limε↓0

ˆ

Rn\B(x,ε)

xj − yj|x− y|n+1

f(y) dy,

is also important in harmonic analysis. Importantly, if w ∈ Ap, thenˆ

Rn

|Rjf(x)|pw(x) dx ≤ C

ˆ

Rn

|f(x)|pw(x) dx

for all f ∈ L∞c .

Recently, more and more Banach lattices come into play in harmonic analysis. Forexample, local Morrey spaces play important role to describe the real interpolation ofweighted Lebesgue spaces, which covers the off-range of the Stein–Weiss interpolationtheorem [3].

Let 1 ≤ q ≤ p < ∞. The local Morrey norm ‖ · ‖LMpq

is given by

‖f‖LMpq≡ sup

R>0|[−R,R]n|

1p− 1

q ‖fχ[−R,R]n‖Lq

for a measurable function f . The local Morrey space LMpq is the set of all the

measurable functions f for which the norm ‖f‖LMpq

is finite. Following the notationin the works [4, 10, 49], we define weighted local Morrey spaces as follows: For ameasurable function f and the weights u and w, we write

‖f‖LMpq(u,w) ≡ sup

R>0u([−R,R]n)

1p− 1

q ‖fχ[−R,R]n‖Lq(w).

The two-weight local Morrey space LMpq(u, w) is the set of all measurable functions

f for which the norm ‖f‖LMpq(u,w) is finite. If u = 1, then we call LMp

q(1, w) thelocal Morrey space of Samko type based on [47, 48] and if u = w, then we callLMp

q(w,w) the local Morrey space of Komori–Shirai type based on [37]. When p = q,LMp

p(u, w) = Lp(w) and hence LMpp(1, w) = LMp

p(w,w) = Lp(w) with coincidenceof norms. So, in this case the theory of Ap applies readily.

The weighted local Morrey space LMpq(u, w) is a contrast of the weighted Morrey

space Mpq(u, w) which consists of all measurable functions f for which the norm

‖f‖Mpq(u,w) ≡ sup

Q∈Qu(Q)

1p− 1

q ‖fχQ‖Lp(w)

is finite. If u(x) = |x|β and w(x) = 1 or u(x) = v(x) = |x|β with β ∈ R, then we saythat LMp

q(u, w) is the power weighted local Morrey space and that Mpq(u, w) is the

power weighted Morrey space.In this paper, assuming that 1 < q < p < ∞, we seek a characterization for the

Hardy–Littlewood maximal operator M to be bounded mainly on LMpq(1, w) and

LMpq(w,w), motivated by the characterization due to Muckenhoupt.

For ν ∈ Z and m = (m1, m2, . . . , mn) ∈ Zn, we define Qνm ≡

∏nj=1

[

mj

2ν,mj+1

2ν

)

.

Denote by D = D(Rn) the set of such cubes. The elements in D are called dyadiccubes. Denote by dist∞ the ℓ∞-distance on R

n. We define the base LQ of cubes by

LQ ≡ Q ∈ D : dist∞(0, Q) = ℓ(Q).

Weighted local Morrey spaces 69

We notice that the dyadic cubes in LQ are pairwise disjoint and that⋃

Q : Q ∈ LQ = Rn \ 0.

We also notice that the number of dyadic cubes in LQ with the same side length is2n(2n − 1).

We define the weight class G by the set of all weights u that satisfy the following:

(i) u is doubling and reverse doubling at the origin, that is, there exist β > α > 1such that

αu([−R,R]n) ≤ u([−2R, 2R]n) ≤ βu([−R,R]n), R > 0;

(ii) u is doubling with respect to LQ in the following sense: there exists γ > 1such that

u([−2ℓ(Q), 2ℓ(Q)]n) ≤ γu(Q)

for all Q ∈ LQ.

Our results are besed upon the following structure of weighted local Morreyspaces.

Lemma 1.1. Let 1 < q < p < ∞ and let u and w be weights. Assume that

u ∈ G. Then

‖f‖LMpq(u,w) ∼ sup

Q∈LQu(Q)

1p− 1

q ‖fχQ‖Lq(w)

holds for any measurable function f .

Lemma 1.1 will be proved in Section 2.Let 1 < q < p < ∞, and let u and w be weights. For a measurable function g its

LMpq(u, w)-associate norm ‖g‖LMp

q(u,w)′ is defined by

‖g‖LMpq(u,w)′ ≡ sup

‖f · g‖L1 : f ∈ LMpq(u, w), ‖f‖LMp

q(u,w) ≤ 1

.

The space LMpq(u, w)

′ collects all measurable functions g for which the norm‖g‖LMp

q(u,w)′ is finite. The space LMpq(u, w)

′ is called the Köthe dual of LMpq(u, w)

or the associated space of LMpq(u, w). Below, we list our results.

Theorem 1.2. Let 1 < q < p < ∞ and let u and w be weights. Assume that

u ∈ G. Then the following are equivalent:

(1) The Hardy–Littlewood maximal operator M is bounded on LMpq(u, w);

(2) there exists a constant C which is independent of Q ∈ LQ such that

supR∈Q, R⊂2Q

1

|R|‖χR‖Lq(w)‖χR‖Lq(w)′ + sup

R∈Q, R⊃Q

1

|R|‖χQ‖LMp

q(u,w)‖χR‖LMpq(u,w)′ ≤ C.

As a special case of Theorem 1.2, we obtain the following characterizations ofLMp

q(1, w) and LMpq(w,w).

Corollary 1.3. Let 1 < q < p < ∞ and let w be a weight. Then the following

are equivalent:

(1) The Hardy–Littlewood maximal operator M is bounded on LMpq(1, w);


supR∈Q, R⊂2Q

1


R∈Q, R⊃Q

1

|R|‖χQ‖LMp

q(1,w)‖χR‖LMpq(1,w)′ ≤ C.

Proposition 1.4. Let 1 < q < p < ∞ and let w be a weight. Then the following

are equivalent:

(1) The Hardy–Littlewood maximal operator M is bounded on LMpq(w,w);

(2) w ∈ G and there exists a constant C which is independent of Q ∈ LQ such

that

supR∈Q, R⊂2Q

1


R∈Q, R⊃Q

1

|R|‖χQ‖LMp

q(w,w)‖χR‖LMpq(w,w)′ ≤ C.

Our characterization can be applied to singular integral operators including theRiesz transform. A singular integral operator is an L2-bounded linear operator Tthat comes with a function K : Rn ×R

n → C satisfying the following conditions:

(1) (Size condition); For all x, y ∈ Rn,

(1.1) |K(x, y)| . |x− y|−n.

(2) (Gradient condition); For all x, y, z ∈ Rn satisfying |x− z| > 2|y − z|,

(1.2) |K(x, z)−K(y, z)| . |x− y|−n−1|y − z|.

(3) Let f be an L2-function. For almost all x /∈ supp(f),

(1.3) Tf(x) =

ˆ

Rn

K(x, y)f(y) dy.

The function K is called the integral kernel of T .

Here and below in this paper we use the following notation: Let A,B ≥ 0. ThenA . B and B & A mean that there exists a constant C > 0 such that A ≤ CB,where C depends only on the parameters of importance. The symbol A ∼ B meansthat A . B and B . A happen simultaneously.

For the singular integral operators, we have the following characterization.

Theorem 1.5. Assume that ‖χ2k+1Q0‖LMp

q(u,w) ≥ α‖χ2kQ0‖LMp

q(u,w), Q0 ∈ LQfor some α > 1 independent on k ∈ N. Let T be a singular integral operator. Let

1 < q < p < ∞ and let u and w be weights. Assume that u ∈ G and that there exists

a constant C which is independent of Q ∈ LQ such that

supR∈Q, R⊂2Q

1


R∈Q, R⊃Q

1

|R|‖χQ‖LMp


Then ‖Tf‖LMpq(u,w) . ‖f‖LMp

q(u,w) for all f ∈ L∞c .

The condition on u and w corresponds to the integral condition considered in[44].

Theorem 1.6. Let 1 < q < p < ∞ and let w be a weight. Assume that u ∈ Gand that ‖Rjf‖LMp

q(u,w) . ‖f‖LMpq(u,w) for all f ∈ L∞

c and for all j = 1, 2, . . . , n.

Then there exists a constant C which is independent of Q ∈ LQ such that

supR∈Q, R⊂2Q

1


R∈Q, R⊃Q

1

|R|‖χQ‖LMp


To investigate weighted Morrey spaces of Samko type, we simply let u = 1in Theorems 1.5 and 1.6. Meanwhile, we have investigated the sufficiency of theboundedness of the singular integral operators on weighted Morrey spaces of Komori–Shirai type. However the necessity is somewhat non-trivial. So, we formulate it.

Proposition 1.7. Let 1 < q < p < ∞, and let w be a doubling weight. As-

sume in addition that ‖Rjf‖LMpq(w,w) . ‖f‖LMp

q(w,w) for all f ∈ L∞c and for all

j = 1, 2, . . . , n. Then there exists a constant C which is independent of Q ∈ LQ such

that

supR∈Q, R⊂2Q

1


R∈Q, R⊃Q

1

|R|‖χQ‖LMp

q(w,w)‖χR‖LMpq(w,w)′ ≤ C.

We apply our results to special cases.

Proposition 1.8. Let 1 < q < p < ∞ and let w(x) = wβ(x) = |x|β with β ∈ R.

Then the following are equivalent:

(1) The maximal operator M is bounded on LMpq(1, w);

(2) − q

pn ≤ β < nq

(

1− 1p

)

.

Proposition 1.9. Let 1 < q < p < ∞ and let w(x) = wβ(x) = |x|β with β ∈ R.

Then the following are equivalent:

(1) The maximal operator M is bounded on LMpq(w,w);

(2) −n < β < n(p− 1);(3) w ∈ Ap.

The ranges obtained in Propositions 1.8 and 1.9 are the same as that for weightedMorrey spaces of Komori–Shirai type and Samko type, respectively.

Proposition 1.10. [53] Let 1 < q < p < ∞ and let w(x) = wβ(x) = |x|β with

β ∈ R. Then the following are equivalent:

(1) The maximal operator M is bounded on Mpq(1, w);

(2) −nq

p≤ β < nq

(

1−1

p

)

.

One of the ways to investigate the boundedness of the operators acting on Morreyspaces is to combine the translation and the boundedness of the operators acting oncorresponding local Morrey spaces. Propositions 1.8 and 1.10 are significant in thatProposition 1.10 can not be obtained by the translation of Proposition 1.8.

In [33] Iida and the first author obtained a complete characterization of the dualinequality of Stein type in weighted Morrey spaces Mp

q(1, w) of Samko type. See also[30]. Despite the recent works [44, 46, 53] a complete characterization of the classfor which M is bounded on Mp

q(1, w) or Mpq(w,w) is still missing.

Proposition 1.11. Let 1 < q < p < ∞ and let w(x) = wβ(x) = |x|β with

β ∈ R. Then the following are equivalent:

(1) The maximal operator M is bounded on Mpq(w,w);

(2) −n < β < n(p− 1);(3) w ∈ Ap.

We can consider the weighted norm inequalities for other operators. Let Iα bethe fractional integral operator given by

(1.4) Iαf(x) ≡

ˆ

Rn

f(y)

|x− y|n−αdy, x ∈ R

n,

for a nonnegative measurable function f . The boundedness of the fractional integraloperator can be characterized in a similar way. Let 1 0 satisfying

‖Iαf · w‖Ls ≤ C‖f · w‖Lp

for any nonnegative measurable function f .

To extend this boundedness to Morrey spaces, we can consider two types ofboundedness. One is due to Spanne [52] and the other is due to Adams [1]. Thefollowing theorem corresponds to the result due to Spanne.

Theorem 1.12. Let 1 < q 0 such that

‖Iαf‖LMst (u,w

t) ≤ C‖f‖LMpq(u,wq)

for any nonnegative measurable function f ;

(2) there exists a constant C > 0 such that

‖Mαf‖LMst (u,w

t) ≤ C‖f‖LMpq(u,wq)

for any nonnegative measurable function f ;


supR∈Q, R⊂2Q

ℓ(R)α

|R|‖χR‖Lt(wt)‖χR‖Lq(wq)′ ≤ C

and

supR∈Q, R⊃Q

ℓ(R)α

|R|‖χQ‖LMs

t (u,wt)‖χR‖LMp

q(u,wq)′ ≤ C.

Then;

• (1) implies (2) and (3).• (2) and (3) are equivalent.

• If there exists κ > 1 such that 2‖χQ‖LMst (u,w

t) ≤ κ‖χκQ‖LMst (u,w

t) for all cubes

Q ∈ LQ, then (2) or (3) implies (1).

In the case of power weighted Morrey spaces of Samko type, we have the followingcharacterization of the boundedness.

Proposition 1.13. Let 1 < q < p < ∞ and 1 < t < s < ∞. Assume that

1

s=

1

p−

α

n,

1

t=

1

q−

α

n.

Then for the power weight w(x) = wβ(x) = |x|β with β ∈ R the following are

equivalent:

(1) The fractional maximal operator Mα is bounded from LMpq(1, w

q) to LMst (1,

wt);

(2) −n

s≤ β <

n

p′, that is, β =

n

sor Iα : L

p(wp) → Ls(ws).

We can replace Mα by Iα once we exclude the case of β = −ns

in (2).

In the case of power weighted Morrey spaces of Komori–Shirai type, we have thefollowing well-known characterization of the boundedness.

Proposition 1.14. Let 0 < α < n, β ∈ R, 1 < q < p < ∞ and 1 < t < s < ∞.

Assume that1

s=

1

p−

α

n,

1

t=

1

s−

α

n.


equivalent:

(1) The fractional integral operator Iα is bounded from LMpq(w

t, wq) to LMst (w

t,wt);

(2) −n

t< β <

ns

p′t.

We can replace Iα by Mα once we include the case of β = −nt

in (2).

It may be interesting to compare these propositions with the following knownresults for Lebesgue spaces, the case where p = q and hence s = t in Propositions 1.13and 1.14.

Proposition 1.15. Let 1 0 such that

‖Iαf · wβ‖Ls ≤ C‖f · wβ‖Lp

for all f ∈ L∞c .

(2) −n

s< β <

n

p′.

We can replace Iα by Mα in the above.

As for weighted Morrey spaces of Samko type, we have the following conclusion:

Proposition 1.16. [46, Proposition 4.1] Let 1 < q < p < ∞, 0 < α < n and

1 < t < s < ∞. Assume that

1

s=

1

p−

α

n,

q

p=

t

s.


equivalent:

(1) The fractional maximal operator Mα is bounded from Mpq(1, w

q) to Mst (1, w

t);

(2) −n

s≤ β <

n

p′.

Proposition 1.17. [46, Proposition 4.2] Let 1 < q < p < ∞, 0 < α < n and

1 < t < s < ∞. Assume that

1

s=

1

p−

α

n,

q

p=

t

s.


equivalent:

(1) The fractional integral operator Iα is bounded from Mpq(1, w

q) to Mst (1, w

t);

(2) −n

s< β <

n

p′.

Proposition 1.18. Let 1 < q −

n

t.


equivalent:

(1) The fractional integral operator Iα is bounded from Mpq(w

t, wq) to Mst(w

t, wt);

(2) −n

t< β <

ns

p′t.

We can replace Iα by Mα once we include the case of β = −nt

in (2).

We remark that in the case of radial functions, Komori and Sato obtained thefollowing result:

Proposition 1.19. [36] Let 1 < p < s < ∞, 1 < q ≤ t < ∞ and 0 < α < nsatisfy

1

s=

1

p−

α

n,

1

t=

1

q−

α

n.

Let β satisfy

α−n

q+

n

p< β <

n

q′.

Then for the power weight w(x) = wβ(x) = |x|β with β ∈ R,

‖Iαf · wβ‖LMst≤ C‖f · wβ‖LMp

q

for all radial functions f ∈ L∞c .

There is a huge amount of literatures dealing with weighted Morrey spaces to-gether with their variants. Many researchers investigated the boundedness propertiesof the linear operators acting on weighted Morrey spaces. such as sublinear operator[4, 12, 15, 35], singular integral operators [14, 35, 63], commutators [17, 12, 35, 59, 61],pseudo-differential operators [26], the square functions [11], Toeplitz operators [56],the fractional integral operators [12, 28, 31, 32] and fractional integrals associatedto operators [51, 54, 55] including the related commutators. Applications to partialdifferential equations can be found in [8, 19, 50]. Embedding relations together withthe envelope are investigated in [22]. A passage to the metric measure spaces isdone in [62]. Ye and Wang used the cube testing to get a characterization of a suffi-cient condition which guarantees the boundedness of the Hardy–Littlewood maximaloperator [60]. See [57, 58] for Campanato spaces of Komori–Shirai type.

As for weighted Morrey spaces of Samko type, the boundedness property ofthe sharp maximal operator, the maximal operator, the singular integral opera-tors, the fractional operarots including the multilinear setting are investigated in[18, 29, 44, 45, 46]. we can find its application to singular integral equations in[41]. Liu considered the boundedness of the pseudo-differential operators in the set-ting of generalized Morrey spaces [39]. There are many attempts of obtaining anecessary and sufficient condition for the weighted norm inequality. See [34] for acharacterization of a sufficient condition which guarantees the boundedness of theHardy–Littlewood maximal operator.

The two-weighted Morrey spaces of the type Mpq(u, v) can be found in [24, 43, 49]

including generalized Morrey spaces [5, 9, 12, 13, 20, 21, 25] and their closed subspaces[2].

The remaining part of this paper is organized as follows: In Section 2 we geta characterization of the local Morrey norm ‖ · ‖LMp

q. Based on the observation in

Section 3, we prove Theorem 1.2 in Section 4. Theorems 1.5 and 1.6 are proved inSection 5. Finally, as apply our results for the power weight w(x) = |x|α in Section6.

2. Preliminaries

2.1. Structure of weighted local Morrey spaces–the proof of Lemma 1.1.

We prove Lemma 1.1.

Proof of Lemma 1.1. The proof consists of two auxiliary equivalences.

We first claim

(2.1) ‖f‖LMpq(u,w) ∼ sup

m∈Z(u([−2m+1, 2m+1]n)

1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w).

It is easy to see that

‖f‖LMpq(u,w) ≥ sup

m∈Zu([−2m+1, 2m+1]n)

1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w).

To obtain the reverse inequality, we fix R > 0. Then if we set j = [1 + log2R], wehave

u([−R,R]n)1p− 1

q ‖fχ[−R,R]n‖Lq(w) . u([−2j , 2j]n)1p− 1

q ‖fχ[−2j ,2j ]n‖Lq(w)

thanks to the doubling property of u at the origin. Since q < p and u is reversedoubling at the origin, we have

u([−2j , 2j]n)1p− 1

q

j−1∑

l=−∞

u([−2l, 2l]n)−1p+ 1

q ∼ 1

and hence

u([−2j, 2j]n)1p− 1

q ‖fχ[−2j,2j ]n‖Lq(w)

≤ u([−2j, 2j]n)1p− 1

q

j−1∑

l=−∞

‖fχ[−2l+1,2l+1]n\[−2l,2l]n‖Lq(w)

≤ u([−2j, 2j]n)1p− 1

q

×

j−1∑

l=−∞

u([−2l, 2l]n)−1p+ 1

q · u([−2l, 2l]n)1p− 1

q ‖fχ[−2l+1,2l+1]n\[−2l,2l]n‖Lq(w)

. u([−2j, 2j]n)1p− 1

q

×

j−1∑

l=−∞

u([−2l, 2l]n)−1p+ 1

q supm∈Z

u([−2m+1, 2m+1]n)1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w)

. supm∈Z

u([−2m+1, 2m+1]n)1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w).

Thus,

‖f‖LMpq(u,w) . sup

m∈Zu([−2m+1, 2m+1]n)

1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w),

which yields (2.1).Next, we shall verify that

supm∈Z

u([−2m+1, 2m+1]n)1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w)

∼ supQ∈LQ

u(Q)1p− 1

q ‖fχQ‖Lq(w).(2.2)

A simple geometric observation shows that

u([−2m+1, 2m+1]n)1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w)

≤ 2n(2n − 1) supQ∈LQ, ℓ(Q)=2m

u(Q)1p− 1

q ‖fχQ‖Lq(w).

Thanks to the doubling condition of u (with respect to LQ), we have reverse inequal-ity

u(Q)1p− 1

q ‖fχQ‖Lq(w) . u([−2m+1, 2m+1]n)1p− 1

q ‖fχ[−2m+1,2m+1]n\[−2m,2m]n‖Lq(w)

for all cubes Q ∈ LQ with ℓ(Q) = 2m. Thus, (2.2) is verified and the proof of thelemma is completed.

2.2. The Lerner–Hytönen decomposition. To investigate the boundednessproperty of the singular integral operators in Theorem 1.5, we need the Lerner–Hytönen decomposition.

A collection Qkjk∈N0, j∈Jk of dyadic cubes is said to be sparse, if the union

Ωk ≡⋃

j∈Jk

Qkj , k = 1, 2, . . .

satisfies;

(2.3) χΩk+1=

∑

j∈Jk+1

χQk+1

j≤ χΩk

=∑

j∈Jk

χQkj≤ 1,

and2|Ωk+1 ∩Qk

j | ≤ |Qkj | (j ∈ Jk).

Let Q0 be a cube and f : Q0 → R be a measurable function. Choose mf(Q0) so that

(2.4) |x ∈ Q0 : f(x) > mf (Q0)|, |x ∈ Q0 : f(x) < mf (Q0)| ≤1

2|Q0|.

Note that mf (Q0) is not determined uniquely. The quantity mf (Q0) is called themedian of f over Q0. The mean oscillation of f over a cube Q of level λ ∈ (0, 1) isgiven by;

ωλ(f ;Q) ≡ infc∈C

((f − c)χQ)∗(λ|Q|),

where ∗ denotes the decreasing rearrangement for functions. Hytönen showed thatthere exists a sparse family Qk

jk∈N∪0, j∈Jk ⊂ D(Q0) such that

|f(x)−mf (Q0)| ≤ 2∞∑

k=0

∑

j∈Jk

ω2−n−2(f ;Qkj )χQk

j(x)

for a.e. x ∈ Q0 [27]. See also [38].

Motivated by this, define the distributional dyadic maximal operator M ♯

2−n−2 by

M ♯,d

2−n−2f(x) ≡ supQ∈D(Q0)

χQ(x)ω2−n−2(f ;Q), x ∈ Rn.

3. Hardy–Littlewood maximal function – Proof of

Theorem 1.2 and Proposition 1.4

3.1. Proof of Theorem 1.2. Assume that (1) holds. One can deduce thecondition

1

|R|‖χQ‖LMp

q(u,w)‖χR‖LMpq(u,w)′ . 1

for all dyadic cubes Q ∈ LQ and all cubes R ∈ Q with R ⊃ 2Q in a well-knownmanner: simply use

(

1

|R|

ˆ

R

|f(y)| dy

)

χQ ≤ infx∈Q

Mf(x).


Meanwhile, if Q ∈ LQ, since we are assuming that M is bounded on LMpq(u, w),

u([−4ℓ(Q), 4ℓ(Q)]n)1p− 1

q ‖χ2QM [fχ2Q]‖Lq(w)

. ‖M [fχ2Q]‖LMpq(u,w)

. ‖fχ2Q‖LMpq(u,w)

∼ u([−ℓ(Q)/2, ℓ(Q)/2]n)1p− 1

q ‖fχ2Q‖Lq(w).

Thus, thanks to the doubling property of u at the origin,

‖M [fχ2Q]‖Lq(w) . ‖fχ2Q‖Lq(w).

This is equivalent to

supR∈Q, R⊂2Q

1

|R|‖χR‖Lq(w)‖χR‖Lq(w)′ . 1.

Let us prove the converse: assume that (2) holds. We have that, for Q ∈ LQ,

χQ(x)Mf(x) ≤ χQ(x)M [fχ2Q](x) + C supR∈Q, R⊃2Q

χQ(x)

|R|

ˆ

R

|f(y)| dy.

Together with the Aq-property at Q this implies

u(Q)1p− 1

q ‖χQMf‖Lq(w)

. u(Q)1p− 1

q ‖χQM [fχ2Q]‖Lq(w) + u(Q)1p− 1

q ‖χQ‖Lq(w) supR∈Q, R⊃2Q

1

|R|

ˆ

R

|f(y)| dy

. u([−4ℓ(Q), 4ℓ(Q)]n)1p− 1

q ‖fχ2Q‖Lq(w)

+ ‖χQ‖LMpq(u,w) sup

R∈Q, R⊃2Q

1

|R|‖χR‖LMp

q(u,w)′‖f‖LMpq(u,w)

. ‖f‖LMpq(u,w),

where we have used Lemma 1.1 and the conditions (i) and (ii) of u. Thus, the proofis complete.

3.2. Proof of Proposition 1.4. We need only verify that w ∈ G, if (1) holds.Once this is verified, then we are in the position of using Theorem 1.2.

For Q = [−R,R]n, R > 0, and any measurable set E ⊂ Q, since

|E|

|Q|w(Q)

1p = w(Q)

1p− 1

q

∥

∥

∥

∥

|E|

|Q|χQ

∥

∥

∥

∥

Lq(w)

,

Lp(w) → LMpq(w,w) and M is assumed bounded on LMp

q(w,w), we have that

|E|

|Q|w(Q)

1p ≤ ‖MχE‖LMp

q(w,w) . ‖χE‖LMpq(w,w) ≤ ‖χE‖Lp(w) = w(E)

1p .

This A∞-property of w at the origin is more than enough to guarantee that w ∈ G.

4. Boundedness of the singular integral operators – Proof of

Theorems 1.5, 1.6 and Proposition 1.7

4.1. Proof of Theorem 1.5. Let Q0 ∈ LQ be a fixed cube. Form the Lerner–Hytönen decomposition of Tf at Q0. Then we obtain a sparse family of dyadic cubes


Qkjk∈N0, j∈Jk in Q0 satisfying

|Tf(x)−mTf(Q0)| ≤ 2∞∑

k=0

∑

j∈Jk

ω2−n−2(Tf ;Qkj )χQk

j(x)

for almost every x ∈ Q0. Thus we have

u(Q0)1p− 1

q ‖w1qχQ0Tf‖Lq

≤ 2u(Q0)1p− 1

q

∥

∥

∥

∥

∥

w1q

∞∑

k=0

∑

j∈Jk


j

∥

∥

∥

∥

∥

Lq

+ u(Q0)1p− 1

qw(Q0)1q |mTf(Q0)|.

Let us set σ = w− 1q−1 . To dualize the first term in the right-hand side, we choose a

non-negative function g ∈ Lq′(σ) and consider

I =

ˆ

Rn

∞∑

k=0

∑

j∈Jk

ω(Tf ;Qkj )χQk

j(x)g(x) dx.

Then we have

I =∞∑

k=0

∑

j∈Jk

ˆ

Rn

ω(Tf ;Qkj )χQk

j(x)g(x) dx

=∞∑

k=0

∑

j∈Jk

ω(Tf ;Qkj )

ˆ

Qkj

g(x) dx

≤∞∑

k=0

∑

j∈Jk

ω(Tf ;Qkj )|Q

kj | inf

x∈Qkj

Mg(x).

Let us set Ekj = Qk

j \ Ωk+1. Then we have 2|Ekj | ≥ |Qk

j |. Since T is weak-(1, 1)bounded, we have

I ≤ 2∞∑

k=0

∑

j∈Jk

infx∈Qk

j

Mf(x)|Ekj | inf

x∈Qkj

Mg(x).

Since Ekj k∈N,j∈Jk is a disjoint family contained in Q0, we have

I ≤ 2

ˆ

Q0

Mf(x)Mg(x) dx.

If we use the Lq′(σ)-boundedness of M in Q0 and ‖g‖Lq′(σ) = 1, then we have∥

∥

∥

∥

∥

w1q

∞∑

k=0

∑

j∈Jk


j

∥

∥

∥

∥

∥

Lq

. ‖w1qMf‖Lq(Q0).

Consequently, since M is bounded thanks to Theorem 1.2,

u(Q0)1p− 1

q

∥

∥

∥

∥

∥

w1q

∞∑

k=0

∑

j∈Jk


j

∥

∥

∥

∥

∥

Lq

. u(Q0)1p− 1

q ‖w1qMf‖Lq(Q0)

. ‖Mf‖LMpq(u,w)

. ‖f‖LMpq(u,w).

Since we are assuming that there exists α > 1 such that

‖χ2k+1Q0‖LMp

q(u,w) ≥ α‖χ2kQ0‖LMp

q(u,w)

for all k = 0, 1, 2, . . . and

|mTf(Q0)| .∞∑

l=1

1

|2lQ0|

ˆ

2lQ0

|f(x)| dx,

we have

|mTf(Q)| · u(Q0)1p− 1

qw(Q0)1q .

∞∑

l=1

α−lu(2lQ0)1p− 1

qw(2lQ0)1q

1

|2lQ0|

ˆ

2lQ0

|f(x)| dx

. ‖Mf‖LMpq(u,w)

. ‖f‖LMpq(u,w),

as was to be shown.

4.2. Proof of Theorem 1.6. We say that a sequence Q0, Q1, . . . , QK in LQ isa chain if ℓ(Qk−1) = 2ℓ(Qk) and Qk−1 and Qk intersect at a set of Lebesgue measurezero for all k = 1, 2, . . . , K and Qj ∩Qj = ∅ if |j − k| ≥ 1. We need a lemma.

Lemma 4.1. Let 1 < q < p < ∞ and let w be a weight. Assume that u ∈ Gand that ‖Rjf‖LMp


c and j = 1, 2, . . . , n. Suppose

that we are given a chain Q0, Q1, Q2, Q3 in LQ.

(1) The cubes 2Q0 and Q3 do not intersect.

(2) For any non-negative f ∈ L∞c supported on 2Q0, we have

n∑

j=1

|Rjf(x)| &1

|Q0|

ˆ

2Q0

f(y) dy (x ∈ Q3).

(3) ‖χ2Q0‖LMpq(u,w) ∼ ‖χQ3‖LMp

q(u,w).

(4) There exists a constant C > 0 independent of Q0 ∈ LQ such that

1

|Q0|‖χ2Q0‖Lq(w)‖χ2Q0‖Lq(w)′ ≤ C.

Proof. We suppose that Q0 = [2m, 2m+1)n and Q3 = [2m−3, 2m−2)n for the sakeof simplicity.

(1) A geometric observation shows that 2Q0 = [2m−1, 5 · 2m−1]n and Q3 = [2m−3,2m−2]n, so that 2Q0 and Q3 do not intersect.

(2) Let f ∈ L∞c (2Q0) be a non-negative function. Since

−n

∑

j=1


∼1

ℓ(Q0)n

for any x ∈ Q3 and y ∈ 2Q0, we haven

∑

j=1

|Rjf(x)| = −n

∑

j=1

Rjf(x) = −n

∑

j=1

ˆ

Rn


f(y) dy &1

|Q0|

ˆ

2Q0

f(y) dy.

(3) By considering f = χ2Q0 in (2) and using ‖Rjχ2Q0‖LMpq(u,w) . ‖χ2Q0‖LMp

q(u,w),we can show that ‖χQ3‖LMp

q(u,w) & ‖χ2Q0‖LMpq(u,w). We can swap the role of

2Q0 and Q3 to have ‖χ2Q0‖LMpq(u,w) . ‖χQ3‖LMp

q(u,w) if we go through a similarargument.

(4) From (2) we have

‖χQ0‖LMpq(u,w)

1

|Q0|

ˆ

2Q0

f(y) dy . ‖f‖LMpq(u,w) ∼ u(Q0)

1p− 1

q ‖f‖Lq(w)

for any non-negative measurable function f supported on 2Q0. It remains touse the duality.

Going through a similar argument, we have the following corollary.

Corollary 4.2. Let 1 < q 0 independent of Q ∈ LQ such that

supR∈Q, R⊂2Q

1

|R|‖χR‖Lq(w)‖χR‖Lq(w)′ ≤ C.

Proof. Let R ∈ Q satisfy R ⊂ 2Q. By decomposing R or by expanding R, we canassume that R is a dyadic cube. Then we can choose a sequence S = S1, S2, S3 = R ofcubes such that ℓ(R) ∼ ℓ(Sk) ∼ dist(R, S) for k = 1, 2, 3, that Sk and Sk+1 intersectat a point for k = 1, 2 and that 2S1 ∩ 2S3 = ∅. If we replace Q3 by S and Q0 by R,respectively in the proof of Lemma 4.1, we can argue as before.

We move on to the proof of Theorem 1.6. In view of Corollary 4.2, it remains toshow that there exists a constant C which is independent of Q ∈ LQ such that

supR∈Q, R⊃Q

1

|R|‖χQ‖LMp

q(u,w)‖χR‖LMpq(u,w)′ ≤ C

for all cubes Q ∈ LQ. By expanding R we can assume that R ⊃ 2Q. By decomposingR suitably, we can further assume that there exists a chain Q = Q0, Q1, Q2, Q3 suchthat R and 2Q3 do not intersect.

Then for any non-negative function f ∈ L∞ supported on R, we have

1

|R|

ˆ

R

f(x) dx .

n∑

j=1

|Rjf(y)|,

for all y ∈ Q3 As a result, assuming that ‖Rjf‖LMpq(u,w) . ‖f‖LMp

q(u,w), we see(

1

|R|

ˆ

R

f(x) dx

)

‖χQ3‖LMpq(u,w) . ‖f‖LMp

q(u,w).

Since ‖χQ3‖LMpq(u,w) ∼ ‖χQ‖LMp

q(u,w) thanks to Lemma 4.1(3), we obtain(

1

|R|

ˆ

R

f(x) dx

)

‖χQ‖LMpq(u,w) . ‖f‖LMp

q(u,w).

By passing to the Köthe dual, we obtain ‖χR‖LMpq(u,w)′‖χQ‖LMp

q(u,w) . |R|, as re-quired.

4.3. Proof of Proposition 1.7. We need to prove that w ∈ G in particular, wis reverse doubling at the origin. To this end, we observe

‖Rjf‖LMpq(w,w) . ‖f‖Lp(w)

for all f ∈ L∞c . Let Q be a cube centered at the origin and let Q′ be a cube such that

|Q| = |Q′| and ♯(Q∩Q′) = 1, where ♯E stands for the cardinality of the set E ⊂ Rn.

Then since w is doubling, we have

w(Q)1p1

|Q|

ˆ

Q

|f(y)| dy ∼ w(Q′)1p1

|Q|

ˆ

Q

|f(y)| dy . ‖f‖Lp(w)

for all f ∈ L∞ with supp(f) ⊂ Q. Let R be a cube contained in Q. If we let

f = w− 1qχR, then we obtain w ∈ Aq+1. Thus, we see that w ∈ G and we are in the

position of using Theorem 1.6.

5. Fractional integral operators – Proof of Theorem 1.12

We can prove that (2) and (3) are equivalent similar to Theorem 1.2. Mean-while, (1) is clearly stronger than (2). It remains to show that (3) implies (1) un-der an additional assumption. This is achieved similar to Proposition 1.8 usingM ♯

2−n−2(Iαf)(x) . Mαf(x), (x ∈ Rn).

6. The case of the power weight – Proof of

Propositions 1.8 and 1.9, 1.13 and 1.14

Here we consider the case where w(x) = |x|β where β > −n. Note that w ∈ A∞,so that w ∈ G.

6.1. Proof of Proposition 1.8. Let us assume − q

pn ≤ β < q

(

1− 1p

)

n. It is

clear that

supR∈D, R⊂Q

1

|R|‖χR‖Lq(w)‖χR‖Lq(w)′ . sup

R∈D, R⊂Q

1

|R|‖χRw(cQ)

1q ‖Lq‖χRw(cQ)

− 1q ‖Lq′ = 1

for all cubes Q ∈ LQ. Next we will prove

‖χQ‖Lq(w)‖χR‖LMpq(1,w)′ ≤ C|Q|

1q− 1

p |R|

or equivalently

‖χQ‖LMpq(1,w)‖χR‖LMp

q(1,w)′ . |R|

for all cubes Q ∈ LQ and R such that Q ⊂ R. To this end, by replacing R with alarger one, say 10R, we may assume that R = [−2r, 2r]n is centered at the origin.Write R∗ = [r, 2r]n. By the dilation formula for (LMp

q)′, we have

‖χR∗(2l·)w− 1p‖(LMp

q )′ = 2lβ

q ‖χR∗(2l·)w(2l·)−1q ‖(LMp

q)′ = 2lβ

q− ln

p′ ‖χR∗ · w− 1q ‖(LMp

q )′ .

Then since β < q(

1− 1p

)

n, we have

‖χR‖LMpq(1,w)′ = ‖χRw

− 1q ‖(LMp

q)′ ≤∞∑

l=0

‖χ2−lR\2−l−1Rw− 1

q ‖(LMpq)′

∼∞∑

l=0

‖χR∗(2l·)w− 1q ‖(LMp

q)′ ∼ ‖χR∗w− 1q ‖(LMp

q)′

∼ w(cR∗)−1q ‖χR∗‖(LMp

q)′ .

Thus, since q

pn + β > 0, thanks to [44, Example 2.3]

‖χQ‖Lq(w)‖χR‖LMpq(1,w)′ . w(cQ)

1q ‖χQ‖Lqw(cR∗)−

1q ‖χR∗‖(LMp

q)′

. ℓ(Q)n+β

q w(cR∗)−1q |R|

1p′ . ℓ(Q)

n+β

q ℓ(R)n−n+β

p

. |Q|1q− 1

p |R|.

Let us assume that M is bounded on LMpq(1, |x|

β). Then χ[−1,1]n ∈ LMpq(1, wβ)

and χB |·|−n

log |·|/∈ LMp

q(1, wβ), where B is a small open ball centered at the origin.

We observe that χ[−1,1]n ∈ LMpq(1, wβ) if and only if β ≥ − q

pn and that χB |·|−n

log |·|/∈

LMpq(1, wβ) if and only if β < q(1− 1

p)n.

6.2. Proof of Proposition 1.9. According to [6], (2) and (3) are equivalent.If we assume (1), then M is bounded from Lp(w) to LMp

q(w,w). As a result,

w(Q)1p

(

1

|Q|

ˆ

Q

|f(y)| dy

)

≤ C‖f‖Lp(w)

for all measurable functions f and all cubes centered at the origin. Thus,

w(Q)

|Q|·

(

σ(Q)

|Q|

)p−1

. 1

for all cubes centered at the origin, where σ(x) = w(x)−1

p−1 , x ∈ Rn. As a conse-

quence we have −n < β < n(p− 1).Assume −n < β < n(p− 1). As before, it is clear that

supR∈D, R⊂Q

1

|R|‖χR‖Lq(w)‖χR‖Lq(w)′ ≤ C sup

R∈D, R⊂Q

1

|R|‖χRw(cQ)

1q ‖Lq‖χRw(cQ)

− 1q ‖Lq′ = C

for all cubes Q ∈ LQ.Next we will establish

(6.1) ‖χQ‖Lq(w)‖χR‖LMpq(w,w)′ ≤ Cw(Q)

1q− 1

p |R|

or equivalently‖χQ‖LMp

q(w,w)‖χR‖LMpq(w,w)′ . |R|

for all cubes Q ∈ LQ and R such that Q ⊂ R. To this end, by replacing R with alarger one, say Q(0, 10ℓ(R)), we may assume that R = [−2r, 2r]n is centered at theorigin. Write R∗ = [r, 2r]n. By the dilation formula for LMp

q(w,w)′, we have

‖χR∗(2l·)‖LMpq(w,w)′ = 2−ln+ l(n+β)

p ‖χR∗‖LMpq(w,w)′.

Since w is a power weight

‖χ2−lR\2−l−1R‖LMpq(w,w)′ ∼ ‖χR∗(2l·)‖LMp

q(w,w)′.

Thus,

‖χR‖LMpq(w,w)′ ≤

∞∑

l=0

‖χ2−lR\2−l−1R‖LMpq(w,w)′ .

∞∑

l=0

‖χR∗(2l·)‖LMpq(w,w)′.

Assuming β < n(p− 1), we have

‖χR‖LMpq(w,w)′ .

∞∑

l=0

2−ln+ l(n+β)p ‖χR∗‖LMp

q(w,w)′ ∼ ‖χR∗‖LMpq(w,w)′

. w(R∗)1q− 1

pw(cR∗)−1q |R∗|

1q′ . w(R∗)−

1p |R∗|.

As a result, since n + β > 0, thanks to [44, Example 2.3]

‖χQ‖Lq(w)‖χR‖LMpq(w,w)′ . w(Q)

1qw(R∗)−

1p |R∗| . ℓ(Q)

n+β

q ℓ(R)n−n+β

p

. ℓ(Q)n+β

q−n+β

p |R| . w(Q)1q− 1

p |R|.

Thus (6.1) is proved.

6.3. Proof of Proposition 1.11. (1) is sufficient for (2) and (3) similar toProposition 1.8. Let Q be a cube. We need to show that

(6.2) w(Q)1p− 1

q

(ˆ

Q

Mf(x)qw(x) dx

)1q

. ‖f‖LMpq(w,w)

for all cubes Q. If −n < β < n(q−1), then we can go through the same argument as[37] which is followed by [12, 40, 56, 59] and so on. So we assume β ≥ n(q−1). We mayassume that Q is centered at the origin or that 0 /∈ 4Q. Let ε ∈ (0, q

p(n(p− 1)− β)).

We distinguish four cases:

(1) Assume that f is supported on 3Q and that Q is centered at the origin. Weborrow the idea of [7]. In this case, we have

w(Q) = wβ(Q) ∼ ℓ(Q)n+β

from [44, Example 2.3] and(ˆ

Q

Mf(x)qw(x) dx

)1q

. ℓ(Q)β−n(q−1)+ε

q

(ˆ

Q

Mf(x)q|x|n(q−1)−ε dx

)1q

,

since β ≥ n(q − 1) > n(q − 1)− ε. Since |x|n(q−1)−ε ∈ Aq, we have(ˆ

Q

Mf(x)q|x|n(q−1)−ε dx

)1q

.

(ˆ

3Q

|f(x)|q|x|n(q−1)−ε dx

)1q

.

We next decompose 3Q dyadically to have(ˆ

3Q


)1q

≤∞∑

l=0

(ˆ

3·2−lQ\3·2−l−1Q


)1q

∼∞∑

l=0

(2−lℓ(Q))n(q−1)−ε−β

q

(ˆ

3·2−lQ\3·2−l−1Q

|f(x)|q|x|β dx

)1q

.

∞∑

l=0

(2−lℓ(Q))n(q−1)−ε−β

q (2−lℓ(Q))n+βq

−n+βp ‖f‖Mp

q(1,wβ).

Arithmetic shows that

n(q − 1)− ε− β

q+

n+ β

q−

n + β

p=

n(p− 1)− β

p−

ε

q> 0.

Thus, the series in the most right-hand side converges to have

(6.3) ℓ(Q)n+βp

−n+βq ℓ(Q)

β−n(q−1)+ε

q

(ˆ

3Q


)1q

. ‖f‖Mpq(w,w).

Consequently, we have (6.2).(2) Assume that f is supported outside 3Q and that Q is centered at the origin.

In this case, using [44, Example 2.3] again, we have

w(Q)1p− 1

q

(ˆ

Q

Mf(x)qw(x) dx

)1q

. ℓ(Q)n+β

p supR>ℓ(Q)

1

|Q(R)|

ˆ

Q(R)

|f(y)| dy

. supl∈N

ℓ(Q)n+β

p1

|2lQ|

ˆ

2lQ\Q

|f(y)| dy.

Let l ∈ N be fixed. We decompose 2lQ \Q dyadically to have

ℓ(Q)n+β

p1

|2lQ|

ˆ

2lQ\Q

|f(y)| dy = ℓ(Q)n+β

p

l∑

k=1

1

|2lQ|

ˆ

2kQ\2k−1Q

|f(y)| dy


If we use the Hölder inequality, then we have

ℓ(Q)n+β

p1

|2lQ|

ˆ

2lQ\Q

|f(y)| dy

≤ ℓ(Q)n+β

p

l∑

k=1

2n(k−l)

(

1

|2kQ|

ˆ

2kQ\2k−1Q

|f(y)|q dy

)1q

. ℓ(Q)n+β

p−β

q

l∑

k=1

2n(k−l)−β

qk

(

1

|2kQ|

ˆ

2kQ\2k−1Q

|f(y)|q|y|β dy

)1q

.

l∑

k=1

2n(k−l)−n+β

pk‖f‖Mp

q(w,w).

Sincel

∑

k=1

2n(k−l)−n+β

pk . 1, we obtain (6.2).

(3) Assume that f is supported on 3Q and that 0 /∈ 4Q. Then we have

w(Q)1p− 1

q

(ˆ

Q

Mf(x)qw(x) dx

)1q

∼ (|Q|w(c(Q)))1p− 1

q

(ˆ

Q

Mf(x)qw(c(Q)) dx

)1q

. (|Q|w(c(Q)))1p− 1

q

(ˆ

3Q

|f(x)|qw(c(Q)) dx

)1q

. w(Q)1p− 1

q

(ˆ

3Q

|f(x)|qw(x) dx

)1q

.

(4) Assume that f is supported outside 3Q and that 0 /∈ 4Q. Then we have

w(Q)1p− 1

q

(ˆ

Q

Mf(x)qw(x) dx

)1q

∼ (|Q|w(c(Q)))1p− 1

q

(ˆ

Q

Mf(x)qw(c(Q)) dx

)1q

∼ (|Q|w(c(Q)))1p supR:cubes,R⊃Q

1

|R|

ˆ

R\3Q

|f(x)| dx

∼ (|Q|w(c(Q)))1p supR:cubes,R⊃Q,c(Q)=c(R)

1

|R|

ˆ

R\3Q

|f(x)| dx.

If 0 /∈ 8R, c(Q) = c(R) and Q ⊂ R, then we have

(|Q|w(c(Q)))1p1

|R|

ˆ

R\3Q

|f(x)| dx

. (|R|w(c(R)))1p

1

w(R)

ˆ

R\3Q

|f(x)|w(x) dx

. w(R)1p

(

1

w(R)

ˆ

R\3Q

|f(x)|qw(x) dx

)1q

≤ ‖f‖LMpq(w,w).

Suppose instead that 0 ∈ 8R, c(Q) = c(R) and Q ⊂ R. Then we have

1

|R|

ˆ

R\3Q

|f(x)| dx ≤1

|R|

(ˆ

R


)1q(ˆ

R

|x|−n+ εq−1 dx

)1q′

.1

|R|ℓ(R)

εq

(ˆ

R


)1q

.

From (6.3) we have

(ˆ

R


)1q

. ℓ(R)β−n(q−1)−ε

q w(R)−1p+ 1

q ‖f‖Mpq(w,w).

As a result,

w(Q)1p1

|R|

ˆ

R\3Q

|f(x)| dx . w(R)1p ℓ(R)

β+nq w(R)−

1p+ 1

q ‖f‖Mpq(w,w) . ‖f‖Mp

q(w,w).

Putting these results all together, we obtain the desired result.

6.4. Proof of Proposition 1.13. If we assume (1), then we have χ[−1,1]n ∈LMs

t(1, wt) and χB| · |

−n /∈ LMpq(1, w

q), or equivalently χB| · |−n+β /∈ LMp

q where Bis a small open ball centered at the origin. We observe that χ[−1,1]n ∈ LMs

t (1, wt) if

and only if β ≥ −ns, and that χB| · |

−n+β /∈ LMpq , if and only if β − n < −n

p. If we

assume −ns≤ β < n

p′, then we can argue as we did in the proof of Proposition 1.8,

to conclude that w = wβ satisfies

supR∈Q, R⊂2Q

ℓ(R)β

|R|‖χR‖Lt(wt)‖χR‖Lq(wq)′+ sup

R∈Q, R⊃Q

ℓ(R)β

|R|‖χQ‖LMs

t (1,wt)‖χR‖LMp

q(1,wq)′ ≤ C

for all cubes Q ∈ LQ.In fact, if R is a cube such that R ⊂ 2Q, then we have

ℓ(R)β

|R|‖χR‖Lt(wt)‖χR‖Lq(wq)′ ∼

ℓ(R)β

|R|‖χR‖Lt‖χR‖(Lq)′ = 1.

Meanwhile if R is a cube such that R ⊃ Q, we denote by R∗ the cube satisfying|R∗| = 100n|R| and centered at the origin. Then

ℓ(R)β

|R|‖χQ‖LMs


q(1,wq)′ .ℓ(R∗)β

|R∗||Q|

1s |c(Q)|β‖χR∗‖LMp

q(1,wq)′ .

We note that

‖χR∗‖LMpq(1,wq)′ = ‖w−1χR∗‖(LMp

q )′ ≤ ‖w−1χR∗‖Lp′ . ℓ(R)n

p′−β

,

where we used β < np′

to guarantee that ‖w−1χR∗‖Lp′ = ‖(wβ)−1χR∗‖Lp′ is finite.

Observe also that c(Q) ∼ ℓ(Q). As a result, since −ns≤ β,

ℓ(R)β

|R|‖χQ‖LMs


q(1,wq)′ .ℓ(Q)β+

ns

ℓ(R)β+ns

≤ 1.

We can consider Iα by the use of Theorem 1.12.

6.5. Proof of Proposition 1.14. If we assume (1), then χ[−1,1]n ∈ LMpq(w

t, wq)

and χB|·|−n

log |·|/∈ LMp

q(wt, wq), where as before B is a small open ball centered at

the origin. We observe that χ[−1,1]n ∈ LMpq(w

t, wq) if and only if βt ≥ −n andχB|·|−n

log |·|/∈ LMp

q(wt, wq) if and only if

β <ns

p′t.

If Iα is bounded from LMpq(1, w

q) to LMst(1, w

t), then we can rule out the possibility

of β = −nt, since | · |−

tβ+np

+ tβ−qβq ∈ LMs

t(1, wt). If we assume −n

t< β < ns

p′t, then we

can argue as we did in the proof of Proposition 1.9, that w = wβ satisfy

supR∈Q, R⊂2Q

ℓ(R)β

|R|‖χR‖Lt(wt)‖χR‖Lq(wq)′

+ supR∈Q, R⊃Q

ℓ(R)β

|R|‖χQ‖LMs

t (wt,wt)‖χR‖LMp

q(wt,wq)′ ≤ C

for all cubes Q ∈ LQ. In fact,

supR∈Q, R⊂2Q

ℓ(R)β

|R|‖χR‖Lt(wt)‖χR‖Lq(wq)′

. supR∈Q, R⊂2Q

ℓ(R)βw(cR)

|R|‖χR‖Ltw(cR)

−1‖χR‖(Lq)′ = 1.

We move on to the proof of

supR∈Q, R⊃Q

ℓ(R)β

|R|‖χQ‖LMs


q(wt,wq)′ . 1.

By expanding R suitably, we may assume that R is centered at the origin. Let

λ = β

(

1 +t

p−

t

q

)

+n

p−

n

q+ ε = β

t

s+

n

p−

n

q+ ε.

Here ε > 0 is chosen small enough to have λ < nq′

. This is possible because

n

q′− λ+ ε =

n

p′− β

(

1 +t

p−

t

q

)

>

(

ns

p′t− β

)(

1 +t

p−

t

q

)

> 0.

Meanwhile,

‖χQ‖LMst (w

t,wt) ∼ wt(Q)1s = ℓ(Q)

n+βt

s

and for a non-negative measurable function f

ˆ

R

f(x) dx ≤

(ˆ

R

f(x)q|x|qλ dx

)1q(ˆ

R

|x|−q′λ dx

)1q′

. ℓ(R)n

q′−λ

(ˆ

R

f(x)q|x|qλ dx

)1q

.

We borrow the idea of [7] again. We decompose

(ˆ

R

f(x)q|x|qλ dx

)1q

≤∞∑

j=1

(ˆ

21−jR\2−jR

f(x)q|x|qλ dx

)1q

as usual. We note that(ˆ

R

f(x)q|x|qλ dx

)1q

.

∞∑

j=1

((2−jℓ(R))n+tβ)1q− 1

pwt(2−jR)1p− 1

q

(ˆ

21−jR\2−jR

f(x)q|x|qλ dx

)1q

.

∞∑

j=1

(2−jℓ(R))εwt(2−jR)1p− 1

q

(ˆ

21−jR\2−jR

f(x)q|x|qλ+(n+tβ)(1− q

p)−qε dx

)1q

=

∞∑

j=1

(2−jℓ(R))εwt(2−jR)1p− 1

q

(ˆ

21−jR\2−jR

f(x)q|x|qβ dx

)1q

. ℓ(R)ε‖f‖Mpq(wt,wq).

Thus,

(6.4)

ˆ

R

f(x) dx . ℓ(R)n

q′−λ+ε‖f‖Mp

q(wt,wq).

We note that

β − n+n

q′− λ+ ε = −

n

s− β

t

s.

Consequently,

supR∈Q, R⊃Q

ℓ(R)β

|R|‖χQ‖LMs


q(wt,wq)′ . supR∈Q, R⊃Q

ℓ(Q)ns+β t

s ℓ(R)−ns−β t

s = 1.

Thus, Iβ is bounded from LMpq(w

t, wq) to LMst (w

t, wt) thanks to Theorem 1.12.The proof for Mα is similar.

6.6. Proof of Proposition 1.18. The case of Mα is omitted because this issimilar to the case of Iα. If Iα is bounded from Mp

q(wt, wq) to Ms

t(wt, wt), then we

have −nt< β < ns

p′tas before. Conversely assume −n

t< β < ns

p′t. Let f ∈ L∞

c . If

β ≤ 0, then we can argue as Komori and Shirai did in [37]. Let us assume β > 0.We distinguish four cases as before to show that

(6.5) wt(Q)1s− 1

t

(ˆ

Q

|Iαf(x)|t|x|βt dx

)1t

. ‖f‖Mpq(wt,wq).

(1) If Q is a cube centered at 0 and if f is supported on 3Q, then we have

wt(Q)1s− 1

t

(ˆ

Q

|Iαf(x)|t|x|βt dx

)1t

. wt(Q)1p− 1

q ℓ(Q)β−τ

(ˆ

Q

|Iαf(x)|t|x|τt dx

)1t

.

Here τ satisfies τ ∈

(

t

sβ, β

)

. Since f is supported on 3Q,

wt(Q)1s− 1

t ℓ(Q)β−τ

(ˆ

Q

|Iαf(x)|t|x|τt dx

)1t

. wt(Q)1s− 1

t ℓ(Q)β−τ

(ˆ

3Q

|f(x)|q|x|τq dx

)1q

.


We borrow the idea of [7]. From the definition of the norm,

(ˆ

3Q

|f(x)|q|x|τq dx

)1q

≤∞∑

j=0

(ˆ

3·2−jQ\3·2−j−1Q

|f(x)|q|x|τq dx

)1q

.

∞∑

j=0

(2−jℓ(Q))τ−β

(ˆ

3·2−jQ\3·2−j−1Q

|f(x)|q|x|βq dx

)1q

.

∞∑

j=0

(2−jℓ(Q))τ−βwt(2−jQ)1q− 1

p‖f‖Mpq(wt,wq).

We observe that

τ − β + (n+ tβ)

(

1

q−

1

p

)

= τ + n

(

1

q−

1

p

)

−t

sβ ≥ n

(

1

q−

1

p

)

> 0,

where we used τ > tsβ for the penultimate inequality. Thus, the series is

summable to have (6.5).(2) If Q is a cube centered at 0 and if f is supported outside 3Q, then we have

wt(Q)1s− 1

t

(ˆ

Q

|Iαf(x)|t|x|βt dx

)1t

. wt(Q)1s

ˆ

Rn\3Q

|f(y)|

|y|n−αdy

from the expression of Iαf(x). We note thatˆ

Rn\3Q

|f(y)|

|y|n−αdy

.

∞∑

j=1

1

ℓ(2jQ)n−α

ˆ

2jQ\2j−1Q

|f(y)| dy

.

∞∑

j=1

ℓ(2jQ)α(

1

ℓ(2jQ)n

ˆ

2jQ\2j−1Q

|f(y)|q dy

)1q

.

∞∑

j=1

ℓ(2jQ)α−β+ tqβ

(

1

wt(2jQ)

ˆ

2jQ\2j−1Q

|f(y)|q|y|qβ dy

)1q

.

Since1

q=

1

t+

α

n,

1

p=

1

s+

α

n,

we have

ℓ(2jQ)α−β+ tqβ

wt(2jQ)1p

=ℓ(2jQ)α+

αβt

n

wt(2jQ)1p

=ℓ(2jQ)α+

αβt

n

wt(2jQ)αnwt(2jQ)

1s

∼ wt(2jQ)−1s .

Hence we haveˆ

Rn\3Q

|f(y)|

|y|n−αdy . wt(Q)−

1s‖f‖Mp

q(wt,wq).

Putting these estimates all together, we obtain (6.5).


(3) If Q is a cube such that 0 /∈ 32Q and if f is supported on 3Q, then by theclassical Hardy–Littlewood–Sobolev theorem

wt(Q)1s− 1

t

(ˆ

Q

|Iαf(x)|t|x|βt dx

)1t

. wt(Q)1s− 1

t |c(Q)|β(ˆ

Q

|Iαf(x)|t dx

)1t

. wt(Q)1s− 1

t |c(Q)|β(ˆ

3Q

|f(x)|q dx

)1q

. wt(3Q)1s− 1

t

(ˆ

3Q

|f(x)|q|x|βq dx

)1q

. ‖f‖Mpq(wt,wq).

(4) If Q is a cube such that 0 /∈ 32Q and if f is supported outside 3Q, then

wt(Q)1s− 1

t

(ˆ

Q

|Iαf(x)|t|x|βt dx

)1t

. wt(Q)1s

ˆ

Rn\3Q

|f(y)|

|y − c(Q)|n−αdy

from the integral expression of f . Consequently

wt(Q)1s− 1

t

(ˆ

Q

|Iαf(x)|t|x|βt dx

)1t

. wt(Q)1s

∞∑

j=1

1

ℓ(2jQ)n−α

ˆ

2j+1Q\2jQ

|f(y)| dy.

Let j0 ≥ 6 be the smallest integer such that 0 ∈ 2j0−1Q. Then since |y| ∼ℓ(2j0Q) for all y ∈ 2j0−3Q, we have

j0−4∑

j=1

1

ℓ(2jQ)n−α

ˆ

2j+1Q\2jQ

|f(y)| dy

.

j0−4∑

j=1

ℓ(2jQ)α(

1

ℓ(2jQ)n

ˆ

2j+1Q\2jQ

|f(y)|q dy

)1q

.

j0−4∑

j=1

ℓ(2jQ)αℓ(2j0Q)−β

(

1

ℓ(2jQ)n

ˆ

2j+1Q\2jQ

|f(y)|q|y|qβ dy

)1q

.

j0−4∑

j=1

ℓ(2jQ)αℓ(2j0Q)βt

q−βwt(2jQ)−

1p‖f‖Mp

q(wt,wq).

We note ℓ(2jQ)αwt(2jQ)−1p ∼ ℓ(2jQ)−

ns |c(Q)|−

βtp , since

α−n

p= −

n

s.

As a result,

j0−4∑

j=1

1

ℓ(2jQ)n−α

ˆ

2j+1Q\2jQ

|f(y)| dy

. ℓ(2j0Q)−β+βt

q ℓ(Q)−ns |c(Q)|−

βt

p ‖f‖Mpq(wt,wq)

. ℓ(2j0Q)−β+βt

q−βt

p ℓ(Q)−ns ‖f‖Mp

q(wt,wq)

= ℓ(2j0Q)−βt

s ℓ(Q)−ns ‖f‖Mp

q(wt,wq)

∼ wt(2j0Q)−1s‖f‖Mp

q(wt,wq).

Consequently,

wt(Q)1s

j0−4∑

j=1

1

ℓ(2jQ)n−α

ˆ

2j+1Q\2jQ

|f(y)| dy . ‖f‖Mpq(wt,wq).

We use the idea of [7]. As we did in (6.4), using β < nsp′t

, we have

wt(Q)1s

∞∑

j=j0−3

1

ℓ(2jQ)n−α

ˆ

2j+1Q\2jQ


Thus, putting these observations together, we obtain

wt(Q)1s

∞∑

j=1

1

ℓ(2jQ)n−α

ˆ

2j+1Q\2jQ


All together then, we conclude that Iα is bounded from Mpq(w

t, wq) to Mst(w

t, wt).

Acknowledgement. Shohei Nakamura was partially supported by Grand-in-Aidfor Young Scientist A. 16H05995. Yoshihiro Sawano was partially supported byGrand-in-Aid for Scientific Research (C), No. 16K05209 and 19K03546 for JapanSociety for the Promotion of Science and by People’s Friendship University of Russia.The authors are thankful to Professor Javier Duoandikoetxea for his illuminatingdiscussion of the power weighted Morrey spaces. Especially the authors are motivatedby the paper [7].

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Received 6 November 2018 • Accepted 17 January 2019

weighted local morrey spacesexample, local morrey spaces play important role to describe the real...

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