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,
1Satoko Sugano,
2and Hitoshi Tanaka
31Department 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,yosihiro@math.kyoto-u.ac.jp
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 operatorIα and the classical fractional
maximal operatorMαare given by
Iαfx:
Rn
fy
x−yn1−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,∞
generalized fractional maximal operatorMρby
Tρfx:
Rn
fyρ x−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≡tnα, 0< α <1, then we haveT
ρIαandMρMα.
A well-known fact in partial differential equations is thatIαis an inverse of−Δnα/2.
The operator1−Δ−1admits an expression of the formTρfor 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 inLpRnwe set
f
p,φ:sup
Q∈QφQ
1
|Q|
Q
fxp
dx
1/p
. 1.3
We will call the Morrey spaceMp,φRn Mp,φthe subset of all functionsflocally inLpRn
for whichfMp,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⊂ Mp2,φwhenp
1≥p2>0. We remark that without the loss of generality we may assume
φtis nondecreasing butφtpt−n is nonincreasing. 1.4
See1.Hereafter, we always postulate1.4onφ. Ifφt≡ tn/p0,p
0 ≥p,Mp,φcoincides with the usual Morrey space and we write this forMp,p0and the norm for ·
Mp,p0. Then we have the inclusion
Lp0 Mp0,p0 ⊂ Mp1,p0 ⊂ Mp2,p0 1.5
whenp0≥p1≥p2>0.
In the present paper, we take up some relations between the generalized fractional integral operatorTρand the generalized fractional maximal operatorMρin the framework of
the Morrey spacesMp,φTheorem 1.2. In the last section, we prove a dual version of Olsen’s
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 2n/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≤1andb < a. Suppose thatρtmaxap,bqt−nis nonincreasing. Then
g·Tρfp,ρa ≤Cgq,ρbMρ1−bf
p,ρa, 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 replaceρawithφandρb
withη. We now formulate our main theorems. In the sequel we always assume thatρsatisfies
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 φtpt−n and ηtqt−n are nonincreasing. Assume also that
∞
t
ρsηs sρsφsds≤C
ηt
φt ∀t >0, 1.13
then
g·Tρfp,φ≤Cgq,ηMρ/ηfp,φ, 1.14
where the constantCis independent offandg.
Remark 1.3. Let 0 ≤ b≤ 1 andb < a. Thenφ ρaandη ρb satisfy the assumption1.13.
Indeed,
∞
t
ρsρsb sρsρsads
∞
t
ρsb−a−1ρs s ds
∞
t
d ds
1
b−aρs
b−ads≤ 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
Tρ.
Corollary 1.4. Let1≤p <∞. Suppose that
∞
t
ρs sρsφsds≤
C
φt ∀t >0, 1.16
then
Tρfp,φ≤CMρfp,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
Corollary 1.5. Let
1≤p <∞,
⎧ ⎨ ⎩
p≤q if p1,
p < q if p >1. 1.19
Suppose that
∞
t
ρs
sφsds≤C ρt
φt ∀t >0, 1.20
then
g·Tρf
p,φ≤Cgq,ρMfp,φ. 1.21
In particular, if1< p < q <∞, then
g·Tρf
p,φ≤Cgq,ρfp,φ. 1.22
Here,Mdenotes the Hardy-Littlewood maximal operator defined by
Mfx: sup
x∈Q∈Q
1
|Q|
Q
fydy. 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 φtpt−nandηtqt−nare nonincreasing. Suppose also that
ρt φt
∞
t
ρs
sφsds≤C ηt
φtp/r ∀t >0, 1.24
then
g·Tρfr,φp/r ≤Cgq,ηfp,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. Let1< p≤r <∞. Suppose that
ρt φt
∞
t
ρs sφsds≤
C
then
Tρf
r,φp/r ≤Cfp,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
operatorIα. The result holds by lettingρt≡tnα,φt≡tn/p0,andηt≡tn/q0.
Proposition 1.8see1, Proposition 1.7. Let0 < 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
Mr,r0 ≤CgMq,q0fMp,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 fromMp,p0toMr,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
Iαfxs
dx
1/s
≤CfMp,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
gxIαfxr
dx
1/r
≤CgMq,q0fMp,p0 1.31
p/p0 r/r0 1 the case of the Lebesgue spacescorrespondsso-calledto the Feff eran-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. Let1< p≤p0 <∞,1< s≤s0<∞, and0< α <1. Assume that
p p0
s s0,
1
s0 1
p0 −α. 1.33
1Iα:Mp,p0 → Ms,s0is continuous but not surjective.
2Letϕ∈ Sbe an auxiliary function chosen so thatϕx 1,2≤ |x| ≤4and thatϕx 0,
|x| ≤1,|x| ≥8. Then the norm equivalence
fMp,p0
⎛ ⎝∞
j−∞
22jn−αFϕ2j·∗Iαf
2 ⎞ ⎠
1/2
Mp,p0
1.34
holds forf∈ Mp,p0, whereFdenotes the Fourier transform.
In view of this propositionMs,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 spaceMs,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. Let0 < p < ∞. Suppose thatρ,η, andφare nondecreasing and thatηtpt−nand φtpt−nare nonincreasing. Then
g·Mρfp,φ≤Cgp,ηMρ/ηfp,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
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, mQpf:mQfp
1/p
. 2.1
2.1. Proof of
Theorem 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 operatorTρffollowing the idea of P´erezsee
16:
Tρfx
ν∈Z
2ν−1<|x−y|≤2ν
fyρ x−y x−yn dy
≤C
ν∈Z ρ2ν
2nν
Bx,2ν
fydy
≤C
ν∈Z
Q∈D:Qx,Q2ν
ρQ |Q|
3Q
fydy
C
Q∈D
ρQ |Q|
3Q
fydy·χQx
C
Q∈D
ρQm3Q
f·χQx,
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 thatφtpt−nis nonincreasing, it suffices to show
Q0
gxTρfx
p
dx
1/p
≤Cgq,ηMρ/ηfp,φ|Q0|1/pφQ0−1, 2.3
for all dyadic cubesQ0. Hereafter, we let
D1Q0:{Q∈ D:Q⊂Q0},
D2Q0:{Q∈ D:QQ0}.
Let us define fori1,2
Fix:
Q∈DiQ0
ρQm3Q
fχQx 2.5
and we will estimate
Q0
gxFix
p
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 : 2n 1. For
k1,2, . . .let
Dk:
Q∈D1Q0:mQh>γ0ck
Q. 2.7
Considering the maximal cubes with respect to inclusion, one can write
Dk
j
Qk,j, 2.8
where the cubes{Qk,j} ⊂ D1Q0are nonoverlapping. By virtue of the maximality ofQk,jone has that
γ0ck< mQk,jh≤2
nγ
0ck. 2.9
Let
E0 :Q0\D1, Ek,j :Qk,j\Dk 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:mQh≤γ0c
,
Dk,j :
Q∈ D1Q0:Q⊂Qk,j, γ0ck< mQh≤γ0ck 1
.
2.12
Then
D1Q0:D0∪
k,j
WithLemma 2.1in mind, let us return to the proof ofTheorem 1.2. We need only to verify that
Q0
gxF1xdx≤Cgq,η
Q0
Mρ/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
Cγ0ck 1
Q∈Dk,j
ρQ
3Q
fydy. 2.18
By virtue of the support condition and1.8we have
Q∈Dk,j
ρQ
3Q
fydy
log2Qk,j
ν−∞
ρ2ν
⎛ ⎝
Q∈Dk,j:Q2ν
3Q
fydy
⎞ ⎠
≤C
3Qk,j
fydy
⎛
⎝log2Qk,j
ν−∞
ρ2ν
⎞ ⎠
≤C
3Qk,j
fydy
Qk,j
0
ρs s ds
CρQk,j
3Qk,j
fydy.
If we invoke relations|Qk,j| ≤2|Ek,j|andγ0ck< mQk,jg, then2.17is bounded by
CρQk,j
m3Qk,j
fmQk,j
gEk,j. 2.20
Now that we have from the definition of the Morrey norm
mQk,j
g≤mQq
k,j
g≤gq,ηηQk,j
−1, 2.21
we conclude that
2.17≤Cgq,ηρ
Qk,j
ηQk,j
m3Qk,j
fEk,j≤Cgq,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
ρ3Qk,j
η3Qk,j
m3Qk,j
f≤ inf
y∈Qk,j
Mρ/ηf
y. 2.23
Similarly, we have
Q∈D0
ρQm3Q
f
Q
gxdx≤Cgq,η
E0
Mρ/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
gxF1x p
dx
1/p
≤Cgq,η
Q0
Mρ/ηfxpdx
1/p
, 2.25
by the duality argument. Take a nonnegative functionw∈LpQ0, 1/p 1/p1, satisfying thatwLpQ
01 and that
Q0
gxF1x p
dx
1/p
Q0
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
CρQk,j
m3Qk,j
fmQk,j
gwEk,j. 2.29
Using H ¨older’s inequality, we have
mQk,j
gw≤mQq
k,j
gmQq
k,jw≤gq,ηη
Qk,j
−1
mQq
k,jw. 2.30
These yield
2.29≤Cgq,ηρ
Qk,j
ηQk,j
m3Qk,j
fmQq
k,jwEk,j
≤Cgq,η
Ek,j
Mρ/ηfxMwq
x1/qdx.
2.31
Similarly, we have
Q∈D0
ρQm3Q
f
Q
gxwxdx≤Cgq,η
E0
Mρ/ηfxMwq
x1/qdx. 2.32
Summing up all factors we obtain
2.28≤Cgq,η
Q0
Mρ/ηfxMwq
Another application of H ¨older’s inequality gives us that
2.28≤Cgq,η
Q0
Mρ/ηfxpdx
1/p
Q0
Mwqxp/qdx
1/p
. 2.34
Now thatp> q, the maximal operatorMisLp/q-bounded. As a result we have
2.28≤Cgq,η
Q0
Mρ/ηfxpdx
1/p
Q0
wxpdx
1/p
Cgq,η
Q0
Mρ/ηfxpdx
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 ηQm3Q
f
≤C
Q∈D2Q0
ρQηQ ρQmQ
Mρ/ηf
.
2.36
As a consequence we obtain
mQ
Mρ/ηf
≤mQpMρ/ηf
≤Mρ/ηfp,φφQ−1. 2.37
In view of the definition ofD2, for eachν ∈ Zwithν ≥ 1 log2Q0there exists a unique cube inD2whose length is 2ν. Hence, inserting these estimates, we obtain
F2x≤CMρ/ηfp,φ
Q∈D2Q0
ρQηQ ρQφQ
CMρ/ηfp,φ ∞
ν1 log2Q0
ρ2νη2ν
ρ2νφ2ν
≤CMρ/ηfp,φ
∞
Q0
ρsηs sρsφsds.
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≤CMρ/ηfp,φ
ηQ0
φQ0
2.39
for allx∈Q0. Inserting this pointwise estimate, we obtain
Q0
gxF2x p
dx
1/p
≤CmQp
0
gMρ/ηfp,φηQ0φQ0−1|Q0|1/p
≤Cgq,ηMρ/ηfp,φφQ0−1|Q0|1/p.
2.40
This is our desired inequality.
2.2. Proof of
Theorem 1.6
We need some lemmas.
Lemma 2.2see1, Lemma 2.2. Letp >1. Suppose thatφsatisfies1.4, then
Mfp,φ≤Cfp,φ. 2.41
Lemma 2.3. Let1< p≤r <∞. Suppose thatφsatisfies1.4, then
Mφ1−p/rf
r,φp/r ≤Cfp,φ. 2.42
Proof. Letx∈Rnbe a fixed point. For every cubeQxwe see that
φQ1−p/rmQf≤min
φQ1−p/rMfx, φQ−p/rfp,φ
≤sup
t≥0
mint1−p/rMfx, t−p/rfp,φ
f1p,φ−p/rMfxp/r.
2.43
This implies
It follows fromLemma 2.2that for every cubeQ0
mQr
0
Mφ1−p/rf
≤f1p,φ−p/rmQp
0
Mfp/r≤Cfp,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
gxF1x p
dx
1/p
≤Cgq,η
Q0
Mρ/ηfxpdx
1/p
. 2.47
We will use it withpr
Q0
gxF1x r
dx
1/r
≤Cgq,η
Q0
Mρ/ηfxrdx
1/r
. 2.48
The casei2 It follows that
ρQm3Q
f≤Cfp,φρQ
φQ 2.49
from the H ¨older inequality and the definition of the normfp,φ. As a consequence we have
F2x≤Cfp,φ
Q∈D2Q0
ρQ
φQ ≤Cfp,φ ∞
ν1 log2Q0
ρ2ν
φ2ν
≤Cfp,φ
∞
Q0
ρs
sφsds≤Cfp,φ
ηt φtp/r.
2.50
Here, we have used the doubling condition1.8and the fact thatφis nondecreasing in the third inequality. Hence it follows that
Q0
gxF2x r
dx
1/r
≤CmQr
0
gfp,φ ηQ0 φQ0p/r
|Q0|1/r
≤Cgq,ηfp,φφQ0−p/r|Q0|1/r.
Combining2.48and2.51, we obtain
g·Tρfr,φp/r ≤Cgq,η
Mρ/ηfr,φp/r fp,φ
. 2.52
We note that the assumption1.24impliesρt/ηt ≤ Cφt1−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 operatorTρ 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
mQpb≤ φQ
|Q| . 3.1
The spaceBp,φRn Bp,φ is defined by the set of all functionsflocally inLpRnwith the norm
fBp,φ:inf
{λk}l1:f
k
λkbk
<∞, 3.2
where eachbk is ap, φ-block and{λk}l1 k|λk| < ∞, and the infimum is taken over
all possible decompositions off. Ifφt≡tn/p0,p
0 ≥p,Bp
,φ
is the usual block spaces, which we write forBp,p0 and the norm for ·
Bp,p0, because the right-hand side of3.1is equal to |Q|1/p0−1|Q|−1/p0. It is easy to prove
Lp0 Bp0,p0 ⊃ Bp1,p0 ⊃ Bp2,p0 3.3
when 1< p0≤p1≤p2<∞. In17, Theorem 1and18, Proposition 5, it was established that the predual space ofMp,φisBp,φ. More precisely, ifg ∈ Mp,φ, thenf ∈ Bp,φ→
Rnfxgxdx
is an element ofBp,φ∗. Conversely, any continuous linear functional inBp,φcan be realized
with someg∈ Mp,φ.
Theorem 3.1. Let 1 < p ≤ r < q < ∞. Suppose that φt and ηt are nondecreasing but that φtpt−nandηtqt−nare nonincreasing. Suppose also that
ρt φt
∞
t
ρs
sφsds≤C ηt
then
TρgfBp,φ ≤CgMq,ηfBr,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. Let1 < p <∞. Suppose thatφis nondecreasing but thatφtpt−nis nonincreasing. Suppose also that
ρt φt
∞
t
ρs sφsds≤
C
φtp/r ∀t >0, 3.6
then
Tρf
Bp,φ≤CfBr,φp/r. 3.7
We dare restateCorollary 3.2in terms of the fractional integral operatorIα. The results
hold by lettingρt≡tnα,φt≡tn/p0,ηt≡1,andgx≡1.
Proposition 3.3see1, Proposition 3.8. Let0< α <1,1< p≤p0 <∞, and1< r ≤r0 <∞.
Suppose that1/p0> α,1/r01/p0−α, andr/r0p/p0, then
IαfBp,p0 ≤CfBr,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 forIα.
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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