A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

Volume 2009, Article ID 835865,18pages doi:10.1155/2009/835865

Research Article

A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

Yoshihiro Sawano,

Satoko Sugano,

and Hitoshi Tanaka

1Department of Mathematics, Kyoto University, Kitasir-akawa, Sakyoku, Kyoto 606-8502, Japan

2Kobe City College of Technology, 8-3 Gakuen-higashimachi, Nishi-ku, Kobe 651-2194, Japan

3Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

Correspondence should be addressed to Yoshihiro Sawano,[email protected] Received 21 July 2009; Revised 31 August 2009; Accepted 13 December 2009

Recommended by Peter Bates

We show some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces.

Copyrightq2009 Yoshihiro Sawano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The present paper is an offspring of1. We obtain some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces.

They generalize what was shown in1. We will go through the same argument as1.

For 0 < α < 1 the classical fractional integral operatorI_α and the classical fractional maximal operatorMαare given by

Iαfx:

Rⁿ

f y x−y^n1−αdy, Mαfx: sup

x∈Q∈Q

1

|Q|^1−α

Q

f ydy.

1.1

In the present paper, we generalize the parameter α. Let ρ : 0,∞ → 0,∞ be a suitable function. We define the generalized fractional integral operator T_ρ and the

generalized fractional maximal operatorMρby

T_ρfx:

Rⁿf

yρx−y x−yⁿ dy, Mρfx: sup

x∈Q∈Q

ρQ

|Q|

Q

f ydy.

1.2

Here, we use the notationQto denote the family of all cubes inRⁿwith sides parallel to the coordinate axes,Q, to denote the sidelength ofQand|Q|to denote the volume of Q. Ifρt≡t^nα, 0< α <1, then we haveT_ρI_αandM_ρM_α.

A well-known fact in partial differential equations is thatI_αis an inverse of−Δ^nα/2. The operator1−Δ⁻¹admits an expression of the formT_ρfor someρ. For more details of this operator we refer to2. As we will see, these operators will fall under the scope of our main results.

Among other function spaces, it seems that the Morrey spaces reflect the boundedness properties of the fractional integral operators. To describe the Morrey spaces we recall some definitions and notation. All cubes are assumed to have their sides parallel to the coordinate axes. ForQ∈ Qwe usecQto denote the cube with the same center asQ, but with sidelength ofcQ.|E|denotes the Lebesgue measure ofE⊂Rⁿ.

Let 0< p <∞andφ :0,∞ → 0,∞be a suitable function. For a functionflocally inL^pRⁿwe set

f

p,φ:sup

Q∈QφQ

1

|Q|

Q

fx^pdx _1/p

. 1.3

We will call the Morrey spaceM^p,φRⁿ M^p,φthe subset of all functionsflocally inL^pRⁿ for whichf_M^p,φ fp,φis finite. Applying H ¨older’s inequality to1.3, we see thatfp1,φ ≥ fp2,φprovided thatp₁ ≥p₂>0. This tells us thatM^p1,φ⊂ M^p2,φwhenp₁≥p₂>0. We remark that without the loss of generality we may assume

φtis nondecreasing butφt^pt⁻ⁿ is nonincreasing. 1.4

See1.Hereafter, we always postulate1.4onφ.

Ifφt≡ t^n/p0,p₀ ≥p,M^p,φcoincides with the usual Morrey space and we write this forM^p,p0and the norm for · M^p,p0. Then we have the inclusion

L^p0 M^p0,p0 ⊂ M^p1,p0 ⊂ M^p2,p0 1.5 whenp₀≥p₁≥p₂>0.

In the present paper, we take up some relations between the generalized fractional integral operatorT_ρand the generalized fractional maximal operatorM_ρin the framework of the Morrey spacesM^p,φTheorem 1.2. In the last section, we prove a dual version of Olsen’s inequality on predual of Morrey spacesTheorem 3.1. As a corollaryCorollary 3.2, we have the boundedness properties of the operatorT_ρon predual of Morrey spaces.

Letθ:0,∞ → 0,∞be a function. By the Dini condition we mean thatθfulfills ₁

0

θs

s ds <∞, 1.6

while the doubling condition onθwith a doubling constantC1>0is thatθsatisfies 1

C₁ ≤ θs

θt ≤C₁, if 1 2 ≤ s

t ≤2. 1.7

We notice that1.4is stronger than the doubling condition. More quantitatively, if we assume 1.4, then φ satisfies the doubling condition with the doubling constant 2^n/p. A simple consequence that can be deduced from the doubling condition ofθis that

log 2 C1 θt≤

_t

t/2

θs

s ds≤log 2·C₁θt ∀t >0. 1.8

The key observation made in1is that it is frequently convenient to replaceθsatisfying1.6 and1.7byθ:

θt _t

0

θs

s ds. 1.9

Before we formulate our main results, we recall a typical result obtained in1.

Proposition 1.1see1, Theorem 1.3. Let

1≤p <∞,

⎧⎨

⎩

p≤q if p1,

p < q if p >1, 1.10

0≤b≤1 andb < a. Suppose thatρt^maxap,bqt⁻ⁿis nonincreasing. Then g·T_ρf

p, ρ^a ≤Cg

q, ρ^bM_ρ1−bf

p, ρ^a, 1.11

where the constantCis independent offandg.

The aim of the present paper is to generalize the function spaces to whichf and g belong. With theorem 1.2, which we will present just below, we can replaceρ^awithφandρ^b withη. We now formulate our main theorems. In the sequel we always assume thatρsatisfies 1.6and1.7, andCis used to denote various positive constants.

Theorem 1.2. Let

1≤p <∞,

⎧⎨

⎩

p≤q if p1,

p < q if p >1. 1.12

Suppose that φt and ηt are nondecreasing but that φt^pt⁻ⁿ and ηt^qt⁻ⁿ are nonincreasing.

Assume also that

_∞

t

ρsηs

sρsφsds≤Cηt

φt ∀t >0, 1.13

then

g·T_ρf

p,φ≤Cg

q,ηM_ρ/ηf

p,φ, 1.14

where the constantCis independent offandg.

Remark 1.3. Let 0 ≤ b≤ 1 andb < a. Thenφ ρ^aandη ρ^b satisfy the assumption1.13.

Indeed,

_∞

t

ρsρs^b s ρsρs^ads

_∞

t

ρs^b−a−1ρs s ds

_∞

t

d ds

1

b−aρs^b−a

ds≤ 1

a−bρt^b−a.

1.15

Hence,Theorem 1.2generalizesProposition 1.1.

Lettingηt≡1 andgx≡1 inTheorem 1.2, we obtain the result of howMρcontrols T_ρ.

Corollary 1.4. Let 1≤p <∞. Suppose that _∞

t

ρs

s ρsφsds≤ C

φt ∀t >0, 1.16

then

T_ρf

p,φ≤CM_ρf

p,φ. 1.17

Corollary 1.4 generalizes 3, Theorem 4.2. Letting η ρ in Theorem 1.2, we also obtain the condition ongandρunder which the mapping

f∈ Mp,φ−→g·Tρf ∈ Mp,φ 1.18 is bounded.

Corollary 1.5. Let

1≤p <∞,

⎧⎨

⎩

p≤q if p1,

p < q if p >1. 1.19

Suppose that

_∞

t

ρs

sφsds≤Cρt

φt ∀t >0, 1.20

then

g·Tρf

p,φ≤Cg

q, ρMf

p,φ. 1.21

In particular, if 1< p < q <∞, then g·Tρf

p,φ≤Cg

q,ρf

p,φ. 1.22

Here,Mdenotes the Hardy-Littlewood maximal operator defined by

Mfx: sup

x∈Q∈Q

1

|Q|

Q

f

ydy. 1.23

We will establish thatMis bounded onMp,φwhenp >1Lemma 2.2. Therefore, the second assertion is immediate from the first one.

Theorem 1.6. Let 1 < p ≤ r < q < ∞. Suppose that φt and ηt are nondecreasing but that φt^pt⁻ⁿandηt^qt⁻ⁿare nonincreasing. Suppose also that

ρt φt

_∞

t

ρs

sφsds≤C ηt

φt^p/r ∀t >0, 1.24

then

g·T_ρf

r,φ^p/r ≤Cg

q,ηf

p,φ, 1.25

where the constantCis independent offandg.

Theorem 1.6 extends 4, Theorem 2, 1, Theorem 1.1, and 5, Theorem 1. As the special caseηt≡1 andgx≡1 inTheorem 1.6shows, this theorem covers1, Remark 2.8.

Corollary 1.7see1, Remark 2.8, see also6–8. Let 1< p≤r <∞. Suppose that ρt

φt _∞

t

ρs

sφsds≤ C

φt^p/r ∀t >0, 1.26

then

Tρf

r,φ^p/r ≤Cf

p,φ. 1.27

Nakai generalizedCorollary 1.7to the Orlicz-Morrey spaces9, Theorem 2.2and10, Theorem 7.1.

We dare restate Theorem 1.6 in the special case when Tρ is the fractional integral operatorI_α. The result holds by lettingρt≡t^nα,φt≡t^n/p0,andηt≡t^n/q0.

Proposition 1.8see1, Proposition 1.7. Let 0< α <1, 1< p≤p₀ <∞, 1< q≤q₀ <∞, and 1< r≤r0<∞. Suppose thatq > r, 1/p0 > α, 1/q0 ≤α, 1/r01/q0 1/p0−α, andr/r0p/p0

then

g·Iαf

M^r,r0 ≤Cg

M^q,q0f

M^p,p0, 1.28

where the constantCis independent offandg.

Proposition 1.8extends4, Theorem 2 see1, Remark 1.9.

Remark 1.9. The special case q₀ ∞ and gx ≡ 1 inProposition 1.8 corresponds to the classical theorem due to Adamssee11.

The fractional integral operatorIα, 0< α <1, is bounded fromM^p,p0toM^r,r0if and only if the parameters 1< p≤p₀<∞and 1< r ≤r₀<∞satisfy 1/r₀1/p₀−αandr/r₀p/p₀.

Using naively the Adams theorem and H ¨older’s inequality, one can prove a minor part of q inProposition 1.8. That is, the proof ofProposition 1.8 is fundamental provided p/p0q0≤q≤q₀.Indeed, by virtue of the Adams theorem we have, for any cubeQ∈ Q,

|Q|^1/s0 1

|Q|

Q

Iαfx^sdx _1/s

≤Cf

M^p,p0, 1 s p0

p 1 s0, 1

s0 1

p0 −α. 1.29

The conditionr/r0p/p0, 1/r01/q0 1/p0−αreads 1

r p₀ p

1 q0

1 p0 −α

p₀

p 1 q0

1

s. 1.30

These yield

|Q|^{1/q0 1/s0} 1

|Q|

Q

gxIαfx^rdx _1/r

≤Cg

M^q,q0f

M^p,p0 1.31

ifr/r₀p/p₀q/q₀. In view of inclusion1.5, the same can be said whenp/p0q0≤q≤q₀. Also observe that 1/r0 1/q0 1/p0 −α > 1/q0.Hence we haveq0 > r0. Thus, since the conditionq > r,Proposition 1.8is significant only whenp/p0r0 < q < p/p0q0.The case

p/p0 r/r0 1 the case of the Lebesgue spacescorrespondsso-calledto the Fefferan- Phong inequalitysee12. An inequality of the form

Rⁿ|ux|²vxdx≤Cv

Rⁿ|∇ux|²dx, 0≤u∈C₀^∞Rⁿ, v≥0 1.32 is called the trace inequality and is useful in the analysis of the Schr ¨odinger operators. For example, Kerman and Sawyer utilized an inequality of type1.32to obtain an eigenvalue estimates of the operatorssee13. By lettingα 1/n, we obtain a sharp estimate on the constantCvin1.32.

In 14, we characterized the range of Iα, which motivates us to consider Proposition 1.8.

Proposition 1.10see14. Let 1< p≤p0 <∞, 1< s≤s0<∞, and 0< α <1. Assume that p

p0 s

s0, 1 s0 1

p0 −α. 1.33

1Iα:M^p,p0 → M^s,s0is continuous but not surjective.

2Letϕ∈ Sbe an auxiliary function chosen so thatϕx 1, 2≤ |x| ≤4 and thatϕx 0,

|x| ≤1,|x| ≥8. Then the norm equivalence

f_Mp,p0

⎛

⎝^∞

j−∞

2^2jn−αFϕ2^j·∗I_αf²

⎞

⎠

1/2 M^p,p0

1.34

holds forf∈ M^p,p0, whereFdenotes the Fourier transform.

In view of this propositionM^s,s0is not a good space to describe the boundedness ofIα, although we have1.29. As we have seen by using H ¨older’s inequality inRemark 1.9, if we use the spaceM^s,s0, then we will obtain a result weaker thanProposition 1.8.

Finally it would be interesting to compare Theorem 1.2 with the following Theorem 1.11.

Theorem 1.11. Let 0< p < ∞. Suppose thatρ,η, andφare nondecreasing and thatηt^pt⁻ⁿand φt^pt⁻ⁿare nonincreasing. Then

g·M_ρf

p,φ≤Cg

p,ηM_ρ/ηf

p,φ, 1.35

where the constantCis independent offandg.

Theorem 1.11generalizes1, Theorem 1.7and the proof remains unchanged except some minor modifications caused by our generalization of the function spaces to whichf andgbelong. So, we omit the proof in the present paper.

2. Proof of Theorems

For any 1 < p < ∞we will write pfor the conjugate number defined by 1/p 1/p 1.

Hereafter, for the sake of simplicity, for anyQ∈ Qand 0< p <∞we will write

mQ

f : 1

|Q|

Q

fxdx, m^p_Q f

:mQf^p1/p

. 2.1

2.1. Proof ofTheorem 1.2

First, we will proveTheorem 1.2. Except for some sufficient modifications, the proof of the theorem follows the argument in15. We denote byDthe family of all dyadic cubes inRⁿ. We assume thatfandgare nonnegative, which may be done without any loss of generality thanks to the positivity of the integral kernel. We will denote byBx, rthe ball centered at xand of radiusr. We begin by discretizing the operatorT_ρffollowing the idea of P´erezsee 16:

T_ρfx

ν∈Z

2^ν−1<|x−y|≤2^νf

yρx−y x−yⁿ dy

≤C

ν∈Z

ρ2^ν 2^nν

Bx,2^νf y

dy

≤C

ν∈Z

Q∈D:Qx,Q2^ν

ρQ

|Q|

3Q

f y

dy

C

Q∈D

ρQ

|Q|

3Q

f y

dy·χ_Qx C

Q∈D

ρQm3Q

f

·χ_Qx,

2.2

where we have used the doubling condition ofρfor the first inequality. To proveTheorem 1.2, thanks to the doubling condition of φ, which holds by use of the facts that φt is nondecreasing and thatφt^pt⁻ⁿis nonincreasing, it suffices to show

Q0

gxTρfxp

dx _1/p

≤Cg

q,ηM_ρ/ηf

p,φ|Q0|^1/pφQ0⁻¹, 2.3

for all dyadic cubesQ0. Hereafter, we let

D1Q0:{Q∈ D:Q⊂Q₀},

D2Q0:{Q∈ D:QQ0}. 2.4

Let us define fori1,2

F_ix:

Q∈DiQ0

ρQm3Q

f

χ_Qx 2.5

and we will estimate

Q0

gxFixp

dx 1/p

. 2.6

The case i 1 and p 1 We need the following crucial lemma, the proof of which is straightforward and is omittedsee15,16.

Lemma 2.1. For a nonnegative functionhin L^∞Q0 one letsγ0 : mQ0hand c : 2^{n 1}. For k1,2, . . .let

Dk:

Q∈D1Q0:mQh>γ0c^k

Q. 2.7

Considering the maximal cubes with respect to inclusion, one can write D_k

j

Q_k,j, 2.8

where the cubes{Qk,j} ⊂ D1Q0are nonoverlapping. By virtue of the maximality ofQk,jone has that

γ0c^k< mQk,jh≤2ⁿγ0c^k. 2.9 Let

E₀ :Q₀\D₁, E_k,j :Q_k,j\D_{k 1}. 2.10

Then{E0} ∪ {Ek,j}is a disjoint family of sets which decomposesQ₀and satisfies

|Q0| ≤2|E0|, Qk,j≤2Ek,j. 2.11

Also, one sets

D0:

Q∈ D1Q0:m_Qh≤γ₀c , Dk,j :

Q∈ D1Q0:Q⊂Qk,j, γ0c^k< mQh≤γ0c^{k 1}

. 2.12

Then

D1Q0:D0∪

k,j

Dk,j. 2.13

WithLemma 2.1in mind, let us return to the proof ofTheorem 1.2. We need only to verify that

Q0

gxF1xdx≤Cg

q,η

Q0

M_ρ/ηfxdx. 2.14

Inserting the definition ofF1, we have

Q0

gxF1xdx

Q∈D1Q0

ρQm3Q

f

Q

gxdx. 2.15

Lettingh g, we will applyLemma 2.1to estimate this quantity. Retaining the same notation asLemma 2.1and noticing2.13, we have

Q0

gxF1xdx

Q∈D0

ρQm3Q

f

Q

gxdx

k,j

Q∈Dk,j

ρQm3Q

f

Q

gxdx.

2.16

We first evaluate

Q∈Dk,j

ρQm3Q

f

Q

gxdx. 2.17

It follows from the definition ofDk,jthat2.17is bounded by

Cγ₀c^{k 1}

Q∈Dk,j

ρQ

3Q

f y

dy. 2.18

By virtue of the support condition and1.8we have

Q∈Dk,j

ρQ

3Q

f y

dy

log₂Qk,j ν−∞ ρ2^ν

⎛

⎝

Q∈Dk,j:Q2^ν

3Q

f y

dy

⎞

⎠

≤C

3Qk,j

f y

dy

⎛

⎝^log2Qk,j

ν−∞ ρ2^ν

⎞

⎠

≤C

3Qk,j

f y

dy

_Qk,j

0

ρs s ds

Cρ

Q_k,j

3Qk,j

f y

dy.

2.19

If we invoke relations|Qk,j| ≤2|Ek,j|andγ0c^k< mQk,jg, then2.17is bounded by

Cρ

Qk,j

m3Qk,j

f mQk,j

gEk,j. 2.20

Now that we have from the definition of the Morrey norm

m_Qk,j g

≤m^q_Q

k,j

g

≤g

q,ηη

Q_k,j−1

, 2.21

we conclude that

2.17≤Cg

q,η

ρ

Qk,j

η

Q_k,jm3Qk,j

fEk,j≤Cg

q,η

Ek,j

Mρ/ηfxdx. 2.22

Here, we have used the fact that ρ is nondecreasing, that η satisfies the doubling condition and that

ρ

3Qk,j

η

3Qk,j

m3Qk,j

f

≤ inf

y∈Qk,j

M_ρ/ηf y

. 2.23

Similarly, we have

Q∈D0

ρQm3Q

f

Q

gxdx≤Cg

q,η

E0

M_ρ/ηfxdx. 2.24

Summing up all factors, we obtain2.14, by noticing that {E0} ∪ {Ek,j}is a disjoint family of sets which decomposesQ₀.

The casei1 andp >1 In this case we establish

Q0

gxF1xp

dx _1/p

≤Cg

q,η

Q0

M_ρ/ηfx^pdx _1/p

, 2.25

by the duality argument. Take a nonnegative functionw∈L^pQ0, 1/p 1/p1, satisfying thatw_L^p_Q01 and that

Q0

gxF1x_p dx

1/p

Q0

gxF1xwxdx. 2.26

Lettinghgw, we will applyLemma 2.1to estimation of this quantity. First, we will insert the definition ofF₁,

Q0

gxF1xwxdx

Q∈D1Q0

ρQm3Q

f

Q

gxwxdx

Q∈D0

ρQm3Q

f

Q

gxwxdx

Q∈Dj,k

ρQm3Q

f

Q

gxwxdx.

2.27

First, we evaluate

Q∈Dk,j

ρQm3Q

f

Q

gxwxdx. 2.28

Going through the same argument as the above, we see that2.28is bounded by C ρ

Qk,j

m3Qk,j

f mQk,j

gwEk,j. 2.29

Using H ¨older’s inequality, we have

m_Qk,j gw

≤m^q_Q

k,j

g m^q_Q

k,jw≤g

q,ηη

Q_k,j−1 m^q_Q

k,jw. 2.30

These yield

2.29≤Cg

q,η

ρ

Qk,j

η

Qk,j

m3Qk,j

f m^q_Q

k,jwE_k,j

≤Cg

q,η

Ek,j

M_ρ/ηfxMw^qx^1/qdx.

2.31

Similarly, we have

Q∈D0

ρQm3Q

f

Q

gxwxdx≤Cg

q,η

E0

M_ρ/ηfxMw^qx^1/qdx. 2.32

Summing up all factors we obtain

2.28≤Cg

q,η

Q0

M_ρ/ηfxMw^qx^1/qdx. 2.33

Another application of H ¨older’s inequality gives us that

2.28≤Cg

q,η

Q0

Mρ/ηfx^pdx

_1/p

Q0

Mw^qx^p/qdx _1/p

. 2.34

Now thatp> q, the maximal operatorMisL^p/q-bounded. As a result we have

2.28≤Cg

q,η

Q0

M_ρ/ηfx^pdx

1/p

Q0

wx^pdx 1/p

Cg

q,η

Q0

Mρ/ηfx^pdx _1/p

.

2.35

This is our desired inequality.

The casei2 andp≥1 By a property of the dyadic cubes, for allx∈Q₀we have

F2x

Q∈D2Q0

ρQm3Q

f

Q∈D2Q0

ρQηQ

ρQ·ρQ

ηQm_3Q f

≤C

Q∈D2Q0

ρQηQ

ρQmQ

Mρ/ηf .

2.36

As a consequence we obtain

m_Q

M_ρ/ηf

≤m^p_Q

M_ρ/ηf

≤M_ρ/ηf

p,φφQ⁻¹. 2.37

In view of the definition ofD2, for eachν ∈ Zwithν ≥ 1 log₂Q0there exists a unique cube inD2whose length is 2^ν. Hence, inserting these estimates, we obtain

F2x≤CMρ/ηf

p,φ

Q∈D2Q0

ρQηQ ρQφQ CM_ρ/ηf

p,φ

∞ ν1 log₂Q0

ρ2^νη2^ν ρ2^νφ2^ν

≤CM_ρ/ηf

p,φ

_∞

Q0

ρsηs sρsφsds.

2.38

Here, in the last inequality we have used the doubling condition1.8and the facts thatρ,φ, andηare nondecreasing and thatρandφsatisfy the doubling condition. Thus, we obtain

F₂x≤CM_ρ/ηf

p,φ

ηQ0

φQ0 2.39

for allx∈Q₀. Inserting this pointwise estimate, we obtain

Q0

gxF2x_p dx

1/p

≤Cm^p_Q

0

gMρ/ηf

p,φηQ0φQ0⁻¹|Q0|^1/p

≤Cg

q,ηM_ρ/ηf

p,φφQ0⁻¹|Q0|^1/p.

2.40

This is our desired inequality.

2.2. Proof ofTheorem 1.6 We need some lemmas.

Lemma 2.2see1, Lemma 2.2. Letp >1. Suppose thatφsatisfies1.4, then Mf

p,φ≤Cf

p,φ. 2.41

Lemma 2.3. Let 1< p≤r <∞. Suppose thatφsatisfies1.4, then Mφ^1−p/rf

r,φ^p/r ≤Cf

p,φ. 2.42

Proof. Letx∈Rⁿbe a fixed point. For every cubeQxwe see that

φQ^1−p/rmQf≤min

φQ^1−p/rMfx, φQ^−p/rf

p,φ

≤sup

t≥0 min

t^1−p/rMfx, t^−p/rf

p,φ

f^1−p/r

p,φ Mfx^p/r.

2.43

This implies

M_φ1−p/rfx^r ≤f^r−p

p,φMfx^p. 2.44

It follows fromLemma 2.2that for every cubeQ0

m^r_Q

0

M_φ^1−p/rf

≤f^1−p/r

p,φ m^p_Q

0

Mf_p/r

≤Cf

p,φφQ0^−p/r. 2.45

The desired inequality then follows.

Proof ofTheorem 1.6. We use definition2.5again and will estimate

Q0

gxFix_r dx

1/r

2.46

fori1,2.

The casei1 In the course of the proof ofTheorem 1.2, we have established2.25

Q0

gxF1xp

dx _1/p

≤Cg

q,η

Q0

M_ρ/ηfx^pdx _1/p

. 2.47

We will use it withpr

Q0

gxF1xr

dx _1/r

≤Cg

q,η

Q0

M_ρ/ηfx^rdx _1/r

. 2.48

The casei2 It follows that

ρQm3Q

f

≤Cf

p,φ

ρQ

φQ 2.49

from the H ¨older inequality and the definition of the normfp,φ. As a consequence we have

F₂x≤Cf

p,φ

Q∈D2Q0

ρQ

φQ ≤Cf

p,φ

∞ ν1 log2Q0

ρ2^ν φ2^ν

≤Cf

p,φ

_∞

Q0

ρs

sφsds≤Cf

p,φ

ηt φt^p/r.

2.50

Here, we have used the doubling condition1.8and the fact thatφis nondecreasing in the third inequality. Hence it follows that

Q0

gxF2xr

dx _1/r

≤Cm^r_Q

0

gf

p,φ

ηQ0

φQ0^p/r|Q0|^1/r

≤Cg

q,ηf

p,φφQ0^−p/r|Q0|^1/r.

2.51

Combining2.48and2.51, we obtain g·T_ρf

r,φ^p/r ≤Cg

q,η

M_ρ/ηf

r,φ^p/r f

p,φ

. 2.52

We note that the assumption1.24impliesρt/ηt ≤ Cφt^1−p/r. Hence we arrive at the desired inequality by usingLemma 2.3.

3. A Dual Version of Olsen’s Inequality

In this section, as an application of Theorem 1.6, we consider a dual version of Olsen’s inequality on predual of Morrey spaces Theorem 3.1. As a corollary Corollary 3.2, we have the boundedness properties of the operatorT_ρ on predual of Morrey spaces. We will define the block spaces following17.

Let 1< p <∞and 1/p 1/p1. Suppose thatφsatisfies1.4. We say that a function bonRⁿis ap, φ-block provided thatbis supported on a cubeQ⊂Rⁿand satisfies

m^p_Qb≤ φQ

|Q| . 3.1

The spaceB^p,φRⁿ B^p,φ is defined by the set of all functionsflocally inL^pRⁿwith the norm

f_Bp,φ:inf

{λk}l¹:f

k

λ_kb_k

<∞, 3.2

where eachb_k is ap, φ-block and{λk}l¹

k|λk| < ∞, and the infimum is taken over all possible decompositions off. Ifφt≡t^n/p0,p₀ ≥p,B^p,φis the usual block spaces, which we write forB^p,p0 and the norm for · _Bp,p

0, because the right-hand side of3.1is equal to

|Q|^1/p0−1|Q|^−1/p0. It is easy to prove

L^p0 B^p0,p0 ⊃ B^p1,p0 ⊃ B^p2,p0 3.3 when 1< p₀≤p₁≤p₂<∞. In17, Theorem 1and18, Proposition 5, it was established that the predual space ofM^p,φisB^p,φ. More precisely, ifg ∈ M^p,φ, thenf ∈ B^p,φ→

Rⁿfxgxdx is an element ofB^p,φ∗. Conversely, any continuous linear functional inB^p,φcan be realized with someg∈ M^p,φ.

Theorem 3.1. Let 1 < p ≤ r < q < ∞. Suppose that φt and ηt are nondecreasing but that φt^pt⁻ⁿandηt^qt⁻ⁿare nonincreasing. Suppose also that

ρt φt

_∞

t

ρs

sφsds≤C ηt

φt^p/r ∀t >0, 3.4

then

T_ρgf

B^p,φ ≤Cg

M^q,ηf

B^r,φp/r, 3.5

ifgis a continuous function.

Theorem 3.1 generalizes 1, Theorem 3.1, and its proof is similar to that theorem, hence omitted. As a special case whengx≡1 andηt≡1, we obtain the following.

Corollary 3.2. Let 1< p <∞. Suppose thatφis nondecreasing but thatφt^pt⁻ⁿis nonincreasing.

Suppose also that

ρt φt

_∞

t

ρs

sφsds≤ C

φt^p/r ∀t >0, 3.6

then

Tρf

B^p,φ≤Cf

B^r,φp/r. 3.7

We dare restateCorollary 3.2in terms of the fractional integral operatorI_α. The results hold by lettingρt≡t^nα,φt≡t^n/p0,ηt≡1,andgx≡1.

Proposition 3.3see1, Proposition 3.8. Let 0< α <1, 1< p≤p0 <∞, and 1< r ≤r0 <∞.

Suppose that 1/p₀> α, 1/r₀1/p₀−α, andr/r₀p/p₀, then I_αf

B^p,p0 ≤Cf

B^r,r0. 3.8

Remark 3.4see1, Remark 3.9. InProposition 3.3, ifr/r₀p/p₀is replaced by 1/r1/p− α, then, using the Hardy-Littlewood-Sobolev inequality locally and taking care of the larger scales by the same manner as the proof ofTheorem 3.1, one has a naive bound forI_α.

Acknowledgments

The third author is supported by the Global COE program at Graduate School of Mathematical Sciences, University of Tokyo, and was supported by F ¯ujyukai foundation.

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