Volume 2009, Article ID 503948,20pages doi:10.1155/2009/503948
Research Article
Boundedness of the Maximal,
Potential and Singular Operators in the Generalized Morrey Spaces
Vagif S. Guliyev
1, 21Department of Mathematics, Ahi Evran University, Kirsehir, Turkey
2Institute of Mathematics and Mechanics, Baku, Azerbaijan
Correspondence should be addressed to Vagif S. Guliyev,vagif@guliyev.com Received 12 July 2009; Accepted 22 October 2009
Recommended by Shusen Ding
We consider generalized Morrey spacesMp,ωRn with a general functionωx, rdefining the Morrey-type norm. We find the conditions on the pairω1, ω2which ensures the boundedness of the maximal operator and Calder ´on-Zygmund singular integral operators from one generalized Morrey spaceMp,ω1Rnto anotherMp,ω2Rn, 1< p < ∞, and from the spaceM1,ω1Rnto the weak spaceWM1,ω2Rn. We also prove a Sobolev-Adams typeMp,ω1Rn → Mq,ω2Rn-theorem for the potential operatorsIα. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities onω1, ω2, which do not assume any assumption on monotonicity ofω1, ω2inr. As applications, we establish the boundedness of some Schr ¨odinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse H ¨older class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.
Copyrightq2009 Vagif S. Guliyev. 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
Forx∈Rnandr >0,letBx, rdenote the open ball centered atxof radiusrand Bx, r denote its complement.
Letf ∈Lloc1 Rn. The maximal operatorM, fractional maximal operatorMα,and the Riesz potentialIαare defined by
Mfx sup
t>0
|Bx, t|−1
Bx,t
f ydy, Mαfx sup
t>0
|Bx, t|−1α/n
Bx,t
f
ydy, 0≤α < n,
Iαfx
Rn
f y
x−ydyn−α, 0< α < n,
1.1
where|Bx, t|is the Lebesgue measure of the ballBx, t.
LetTbe a singular integral Calderon-Zygmund operator, briefly a Calderon-Zygmund operator, that is, a linear operator bounded from L2Rn in L2Rn taking all infinitely continuously differentiable functionsfwith compact support to the functionsTf ∈Lloc1 Rn represented by
Tfx
RnK x, y
f y
dy a.e.on suppf. 1.2
HereKx, yis a continuous function away from the diagonal which satisfies the standard estimates; there existc1>0 and 0< ε≤1 such that
K
x, y≤c1x−y−n 1.3
for allx, y∈Rn, x /y, and
K x, y
−K
x, yK y, x
−K
y, x≤c1
|x−x| x−y
ε
x−y−n, 1.4
whenever 2|x−x| ≤ |x−y|. Such operators were introduced in1.
The operatorsM ≡ M0,Mα,Iα,andT play an important role in real and harmonic analysis and applicationssee, e.g.,2,3.
Generalized Morrey spaces of such a kind were studied in4–20. In the present work, we study the boundedness of maximal operatorMand Calder ´on-Zygmund singular integral operatorsTfrom one generalized Morrey spaceMp,ω1to anotherMp,ω2, 1< p <∞, and from the space M1,ω1 to the weak spaceWM1,ω2. Also we study the boundedness of fractional maximal operatorMαand Riesz potential operatorsMαfromMp,ω1 toMq,ω2, 1< p < q <∞, and from the spaceM1,ω1to the weak spaceWM1,ω2, 1< q <∞.
As applications, we establish the boundedness of some Sch ¨odinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse H ¨older class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.
2. Morrey Spaces
In the study of local properties of solutions to of partial differential equations, together with weighted Lebesgue spaces, Morrey spaces Mp,λRn play an important role; see 21, 22.
Introduced by Morrey23in 1938, they are defined by the norm f
Mp,λ :sup
x,r>0
r−λ/pf
LpBx,r, 2.1
where 0≤λ < n,1≤p <∞.
We also denote byWMp,λthe weak Morrey space of all functionsf ∈WLlocp Rnfor which
f
WMp,λ ≡f
WMp,λRn sup
x∈Rn,r>0r−λ/pf
WLpBx,r<∞, 2.2
whereWLpdenotes the weakLp-space.
Chiarenza and Frasca24 studied the boundedness of the maximal operatorM in these spaces. Their results can be summarized as follows.
Theorem 2.1. Let 1≤p <∞and 0≤λ < n. Then forp >1 the operatorMis bounded inMp,λand forp1Mis bounded fromM1,λtoWM1,λ.
The classical result by Hardy-Littlewood-Sobolev states that if 1< p < q <∞, thenIα
is bounded fromLpRntoLqRnif and only ifαn1/p−1/qand forp 1 < q <∞,Iα
is bounded fromL1RntoWLqRnif and only ifα n1−1/q. S. Spannepublished by Peetre25and Adams26studied boundedness of the Riesz potential in Morrey spaces.
Their results can be summarized as follows.
Theorem 2.2Spanne, but published by Peetre25. Let 0< α < n, 1< p < n/α, 0< λ < n−αp.
Set 1/p−1/qα/nandλ/pμ/q. Then there exists a constantC >0 independent offsuch Iαf
Mq,μ ≤Cf
Mp,λ 2.3
for everyf ∈ Mp,λ.
Theorem 2.3Adams26. Let 0 < α < n, 1 < p < n/α, 0 < λ < n−αp,and 1/p−1/q α/n−λ. Then there exists a constantC >0 independent offsuch
Iαf
Mq,λ≤Cf
Mp,λ 2.4
for everyf ∈ Mp,λ.
Recall that, for 0< α < n,
Mαfx≤υα/n−1n Iαfx, 2.5
hence Theorems2.2and2.3also imply boundedness of the fractional maximal operatorMα, wherevnis the volume of the unit ball inRn.
The classical result for Calderon-Zygmund operators states that if 1< p < ∞thenT is bounded fromLpRntoLpRn, and ifp 1 thenT is bounded fromL1RntoWL1Rn see, e.g.,2.
Fazio and Ragusa27studied the boundedness of the Calder ´on-Zygmund singular integral operators in Morrey spaces, and their results imply the following statement for Calder ´on-Zygmund operatorsT.
Theorem 2.4. Let 1≤p <∞, 0< λ < n. Then for 1< p <∞Calder´on-Zygmund singular integral operatorTis bounded inMp,λand forp1 Tis bounded fromM1,λtoWM1,λ.
Note that in the case of the classical Calder ´on-Zygmund singular integral operators Theorem 2.4was proved by Peetre25. Ifλ0, the statement ofTheorem 2.4reduces to the aforementioned result forLpRn.
3. Generalized Morrey Spaces
Everywhere in the sequel the functionsωx, r, ω1x, randω2x, r,used in the body of the paper are nonnegative measurable function onRn×0,∞.
We find it convenient to define the generalized Morrey spaces in the form as follows.
Definition 3.1. Let 1 ≤ p < ∞. The generalized Morrey space Mp,ωRn is defined of all functionsf ∈Llocp Rnby the finite norm
f
Mp,ω sup
x∈Rn,r>0
r−n/p ωx, rf
LpBx,r. 3.1
According to this definition, we recover the spaceMp,λRnunder the choiceωx, r rλ−n/p:
Mp,λRn Mp,ωRn|ωx,rrλ−n/p. 3.2
In4,5,17,18there were obtained sufficient conditions on weightsω1andω2for the boundedness of the singular operatorT fromMp,ω1RntoMp,ω2Rn. In18the following condition was imposed onwx, r:
c−1ωx, r≤ωx, t≤c ωx, r, 3.3
wheneverr ≤ t ≤ 2r, wherec≥ 1 does not depend ont, r and x ∈ Rn, jointly with the condition
∞
r
ωx, tpdt
t ≤C ωx, rp, 3.4
for the maximal or singular operator and the condition ∞
r
tαpωx, tpdt
t ≤C rαpωx, rp 3.5
for potential and fractional maximal operators, whereC>0does not depend onrandx∈ Rn.
Note that integral conditions of type3.4after the paper28of 1956 are often referred to as Bary-Stechkin or Zygmund-Bary-Stechkin conditions; see also29. The classes of almost monotonic functions satisfying such integral conditions were later studied in a number of papers, see30–32and references therein, where the characterization of integral inequalities of such a kind was given in terms of certain lower and upper indices known as Matuszewska- Orlicz indices. Note that in the cited papers the integral inequalities were studied asr → 0.
Such inequalities are also of interest when they allow to impose different conditions asr → 0 andr → ∞; such a case was dealt with in33,34.
In18the following statements were proved.
Theorem 3.218. Let 1≤p <∞andωx, rsatisfy conditions3.3-3.4. Then forp >1 the operatorsMandTare bounded inMp,ωRnand forp1MandTare bounded fromM1,ωRnto WM1,ωRn.
Theorem 3.318. Let 1≤p <∞,0< α <n/p, 1/q1/p−α/nandωx, tsatisfy conditions 3.3and3.5. Then forp > 1 the operatorsMα andIαare bounded fromMp,ωRntoMq,ωRn and forp1MαandIαare bounded fromM1,ωRntoWMq,ωRn.
4. The Maximal Operator in the Spaces M
p,ωR
nTheorem 4.1. Let 1≤p <∞andf∈Llocp Rn. Then forp >1 Mf
LpBx,t≤Ctn/p ∞
t
r−n/p−1f
LpBx,rdr, 4.1
and forp1
Mf
WL1Bx,t≤Ctn ∞
t
r−n−1f
L1Bx,rdr, 4.2
whereCdoes not depend onf,x∈Rnandt >0.
Proof. Let 1< p <∞. We representfas ff1f2, f1
y f
y
χBx,2t y
, f2
y f
y
χBx,2t y
, t >0, 4.3 and have
Mf
LpBx,t≤Mf1
LpBx,tMf2
LpBx,t. 4.4
By boundedness of the operatorMinLpRn, 1< p <∞we obtain Mf1
LpBx,t≤Mf1
LpRn≤Cf1
LpRnCf
LpBx,2t, 4.5
whereCdoes not depend onf. From4.5we have Mf1
LpBx,t≤Ctn/p ∞
2t
r−n/p−1f
LpBx,rdr
≤Ctn/p ∞
t
r−n/p−1f
LpBx,rdr
4.6
easily obtained from the fact that f LpBx,2tis nondecreasing int, so that f LpBx,2ton the right-hand side of4.5is dominated by the right-hand side of4.6.
To estimateMf2, we first prove the following auxiliary inequality:
Bx,t
x−y−nf
ydy≤C ∞
t
s−n/p−1f
LpBx,sds, 0< t <∞. 4.7
To this end, we chooseβ > n/pand proceed as follows:
Bx,t
x−y−nf
ydy≤β
Bx,t
x−y−nβf ydy
∞
|x−y|s−β−1ds β
∞
t
s−β−1ds
{y∈Rn:t≤|x−y|≤s}x−y−nβf ydy
≤C ∞
t
s−β−1f
LpBx,sx−y−nβ
LpBx,sds.
4.8
Forz∈Bx, twe get
Mf2z sup
r>0
|Bz, r|−1
Bz,r
f2
ydy
≤Csup
r≥2t
Bx,2t∩Bz,r
y−z−nf ydy
≤Csup
r≥2t
Bx,2t∩Bz,r
x−y−nf ydy
≤C
Bx,2t
x−y−nf ydy.
4.9
Then by4.7
Mf2z≤C ∞
2t
s−n/p−1f
LpBx,sds
≤C ∞
t
s−n/p−1f
LpBx,sds,
4.10
whereCdoes not depend onx, r. Thus, the functionMf2z, with fixedxandt, is dominated by the expression not depending onz.Then
Mf2
LpBx,t≤C ∞
t
s−n/p−1f
LpBx,sds 1 LpBx,t. 4.11
Since 1 LpBx,t Ctn/p, we then obtain4.1from4.6and4.11.
Letp1. It is obvious that for any ballBBx, r Mf
WL1Bx,t≤Mf1
WL1Bx,tMf2
WL1Bx,t. 4.12
By boundedness of the operatorMfromL1RntoWL1Rnwe have Mf1
WL1Bx,t≤Cf
L1Bx,2t, 4.13 whereCdoes not depend onx,t.
Note that inequality4.11also true in the casep1. Then by4.11, we get inequality 4.2.
Theorem 4.2. Let 1≤p <∞and the functionω1x, randω2x, rsatisfy the condition ∞
t
ω1x, rdr
r ≤C ω2x, t, 4.14
where C does not depend on xand t. Then forp > 1 the maximal operator M is bounded from Mp,ω1RntoMp,ω2Rnand forp1Mis bounded fromM1,ω1RntoWM1,ω2Rn.
Proof. Let 1< p <∞andf∈ Mp,ω1Rn. ByTheorem 4.1we obtain Mf
Mp,ω2 sup
x∈Rn, t>0
ω2−1x, tt−n/pMf
LpBx,t
≤C sup
x∈Rn, t>0
ω2−1x, t ∞
t
r−n/p−1f
LpBx,rdr.
4.15
Hence
MfM
p,ω2 ≤CfM
p,ω1 sup
x∈Rn, t>0
1 ω2x, t
∞
t
ω1x, rdr r
≤Cf
Mp,ω1
4.16
by4.14, which completes the proof for 1< p <∞.
Letp1 andf∈ M1,ω1Rn. ByTheorem 4.1we obtain Mf
WM1,ω2 sup
x∈Rn, t>0
ω−12 x, tt−nMf
WL1Bx,t
≤C sup
x∈Rn, t>0
ω−12 x, t ∞
t
r−n−1f
L1Bx,rdr.
4.17
Hence
MfWM
1,ω2 ≤CfM
1,ω1Rn sup
x∈Rn, t>0
1 ω2x, t
∞
t
ω1x, rdr r
≤Cf
M1,ω1
4.18
by4.14, which completes the proof forp1.
Remark 4.3. Note that Theorems4.1and4.2were proved in4 see also5.Theorem 4.2do not impose the pointwise doubling conditions3.3and3.4. In the caseω1x, r ω2x, r ωx, r,Theorem 4.2is containing the results ofTheorem 3.2.
5. Riesz Potential Operator in the SpacesM
p,ωR
n 5.1. Spanne Type ResultTheorem 5.1. Let 1≤p <∞, 0< α < n/p, 1/q1/p−α/n,andf∈Llocp Rn. Then forp >1 Iαf
LqBx,t ≤Ctn/q ∞
t
r−n/q−1f
LpBx,rdr, 5.1
and forp1
Iαf
WLqBx,t ≤Ctn/q ∞
t
r−n/q−1f
L1Bx,rdr, 5.2
whereCdoes not depend onf,x∈Rnandt >0.
Proof. As in the proof ofTheorem 4.1, we represent functionfin form4.3and have
Iαfx Iαf1x Iαf2x. 5.3
Let 1< p <∞, 0< α < n/p, 1/q1/p−α/n. By boundedness of the operatorIαfrom LpRntoLqRnwe obtain
Iαf1
LqBx,t≤Iαf1
LqRn
≤Cf1
LpRnC f LpBx,2t.
5.4
Then
Iαf1
LqBx,t≤Cf
LpBx,2t, 5.5 where the constantCis independent off.
Taking into account that f
LpBx,2t≤Ctn/q ∞
2t
r−n/q−1f
LpBx,rdr, 5.6
we get
Iαf1
LqBx,t≤Ctn/q ∞
2t
r−n/q−1f
LpBx,rdr. 5.7
When|x−z| ≤t,|z−y| ≥2t,we have1/2|z−y| ≤ |x−y| ≤3/2|z−y|, and therefore Iαf2
LqBx,t≤
Bx,2t
z−yα−nfydy
LqBx,t
≤C
Bx,2t
x−yα−nf
ydyχBx,t
LqRn.
5.8
We chooseβ > n/qand obtain
Bx,2t|x−y|α−nf
ydyβ
Bx,2t
x−yα−nβf
y∞
|x−y|s−β−1ds
dy
β ∞
2t
s−β−1
{y∈Rn:2t≤|x−y|≤s}
x−yα−nβf ydy
ds
≤C ∞
2t
s−β−1f
LpBx,s|x−y|α−nβ
LpBx,sds
≤C ∞
2t
sα−n/p−1f
LpBx,sds.
5.9
Therefore
Iαf2
LqBx,t ≤Ctn/q ∞
2t
s−n/q−1f
LpBx,sds, 5.10
which together with5.7yields5.1.
Letp1. It is obvious that for any ballBBx, r Iαf
WL1Bx,t≤Iαf1
WL1Bx,tIαf2
WL1Bx,t. 5.11
By boundedness of the operatorIαfromL1RntoWLqRnwe have Iαf1
WL1Bx,t ≤Cf
LqBx,2t, 5.12 whereCdoes not depend onx,t.
Note that inequality5.10also true in the casep1. Then by5.10, we get inequality 5.2.
Theorem 5.2. Let 1 ≤ p < ∞, 0 < α < n/p, 1/q 1/p−α/nand the functions ω1x, rand ω2x, rfulfill the condition
∞
r
tαω1x, tdt
t ≤C ω2x, r, 5.13 where Cdoes not depend onx and r. Then forp > 1 the operatorsMα and Iα are bounded from Mp,ω1RntoMq,ω2Rnand forp1 MαandIαare bounded fromM1,ω1RntoWMq,ω2Rn. Proof. Let 1< p <∞andf∈ Mp,ωRn. ByTheorem 5.1we obtain
Iαf
Mq,ω2 ≤C sup
x∈Rn, t>0
1 ω2x, t
∞
t
r−n/q−1f
LpBx,rdr
≤Cf
Mp,ω1 sup
x∈Rn, t>0
1 ω2x, t
∞
t
rαω1x, rdr r
5.14
by5.13, which completes the proof for 1< p <∞.
Letp1 andf∈ M1,ω1Rn. ByTheorem 5.1we obtain Iαf
WMq,ω2 sup
x∈Rn, t>0
ω2−1x, tt−n/qIαf
WLqBx,t
≤C sup
x∈Rn, t>0
ω2−1x, t ∞
t
r−n/q−1f
L1Bx,rdr.
5.15
Hence
Iαf
WMq,ω2 ≤Cf
M1,ω1Rn sup
x∈Rn, t>0
1 ω2x, t
∞
t
rαω1x, rdr r
≤Cf
M1,ω1
5.16
by5.13, which completes the proof forp1.
Remark 5.3. Note that Theorems5.1and5.2were proved in4 see also5.Theorem 5.2do not impose the pointwise doubling condition,3.3and3.5. In the caseω1x, r ω2x, r ωx, r,Theorem 5.2is containing the results ofTheorem 3.3.
5.2. Adams Type Result
Theorem 5.4. Let 1≤p <∞, 0< α < n/p,andf∈Llocp Rn. Then Iαfx≤Ctα Mfx C
∞
t
rα−n/p−1f
LpBx,rdr, 5.17
whereCdoes not depend onf,x,andt.
Proof. As in the proof ofTheorem 4.1, we represent functionfin form4.3and have
Iαfx Iαf1x Iαf2x. 5.18
ForIαf1x, following Hedberg’s tricksee for instance2, page 354, we obtain|Iαf1x| ≤ C1tαMfx.ForIαf2xwe have
Iαf2x≤
Bx,2t
x−yα−nf ydy
≤C
Bx,2t f
ydy ∞
|x−y|rα−n−1dr
≤C ∞
2t
2t<|x−y|<r
f ydy
rα−n−1dr
≤C ∞
t
rα−n/p−1f
LpBx,rdr,
5.19
which proves5.17.
Theorem 5.5. Let 1≤p <∞, 0< α < n/pand letωx, tsatisfy condition4.14and the conditions
tαωx, t ∞
t
rα ωx, rdr
r ≤Cωx, tp/q, 5.20
whereq≥pandCdoes not depend onx∈Rnandt >0. Suppose also that for almost everyx∈Rn, the functionwx, rfulfills the condition
there exist anaax>0 such thatωx,·:0,∞−→a,∞is surjective. 5.21
Then forp >1 the operatorsMαandIαare bounded fromMp,ωRntoMq,ωp/qRnand forp1 the operatorsMαandIαare bounded fromM1,ωRntoWMq,ω1/qRn.
Proof. In view of the well-known pointwise estimateMαfx≤CIα|f|x, it suffices to treat only the case of the operatorIα.
Let 1≤p <∞andf∈ Mp,ωRn. ByTheorem 5.4we get Iαfx≤Crα Mfx Cf
Mp,ω
∞
r
tαωx, tdt
t . 5.22
From5.20we haverαωx, r ≤Cωx, rp/q. Making also use of condition5.20, we obtain Iαfx≤Cωx, rp/q−1 Mfx Cωx, rp/q f
Mp,ω. 5.23
Sinceωx, ris surjective, we can chooser >0 so thatωx, r Mfx f −1Mp,ωRn, assuming thatfis not identical 0. Hence, for everyx∈Rn, we have
Iαfx≤C
Mfxp/q f1−p/q
Mp,ω . 5.24
Hence the statement of the theorem follows in view of the boundedness of the maximal operatorMinMp,ωRnprovided byTheorem 4.2in virtue of condition4.14
IαfM
q,ωp/q sup
x∈Rn, t>0
ωx, t−p/qt−n/qIαf
LqBx,t
≤Cf1−p/q
Mp,ω sup
x∈Rn, t>0
ωx, t−p/qt−n/qMfp/q
LpBx,t
≤Cf
Mp,ω,
5.25
if 1< p < q <∞and Iαf
WMq,ω1/q sup
x∈Rn, t>0
ωx, t−1/qt−n/qIαf
WLqBx,t
≤Cf1−1/q
M1,ω sup
x∈Rn, t>0
ωx, t−1/qt−n/qMf1/q
WL1Bx,t
≤CfM
1,ω,
5.26
ifp1< q <∞.
6. Singular Operators in the Spaces M
p,ωR
nTheorem 6.1. Let 1≤p <∞andf∈Llocp Rn. Then forp >1 Tf
LpBx,t≤Ctn/p ∞
t
r−n/p−1f
LpBx,rdr, 6.1
and forp1
Tf
WL1Bx,t ≤Ctn ∞
t
r−n−1f
L1Bx,rdr, 6.2
whereCdoes not depend onf,x∈Rnandt >0.
Proof. Let 1< p <∞. We represent functionfas in4.3and have Tf
LpBx,t ≤Tf1
LpBx,tTf2
LpBx,t. 6.3
By boundedness of the operatorT inLpRn, 1 < p < ∞ we obtain Tf1 LpBx,t ≤ Tf1 LpRn≤C f1 LpRn,so that
Tf1
LpBx,t≤Cf
LpBx,2t. 6.4 Taking into account the inequality
f
LpBx,t≤Ctn/p ∞
2t
r−n/p−1f
LpBx,rdr, 6.5
we get
Tf1
LpBx,t≤Ctn/p ∞
2t
r−n/p−1f
LpBx,rdr. 6.6
To estimate Tf2 LpBx,t, we observe that Tf2z≤C
Bx,2t f
ydy
y−zn , 6.7 wherez∈ Bx, tand the inequalities|x−z| ≤ t,|z−y| ≥ 2timply1/2|z−y| ≤ |x−y| ≤ 3/2|z−y|, and therefore
Tf2
LpBx,t≤C
Bx,2t
x−y−nf
ydyχBx,t
LpRn. 6.8
Hence by inequality4.7, we get Tf2
LpBx,t≤Ctn/p ∞
2t
r−n/p−1f
LpBx,rdr. 6.9
From6.6and6.9we arrive at6.1.
Letp1. It is obvious that for any ballBx, r Tf
WL1Bx,t≤Tf1
WL1Bx,tTf2
WL1Bx,t. 6.10
By boundedness of the operatorTfromL1RntoWL1Rnwe have Tf1
WL1Bx,t≤Cf
L1Bx,2t, 6.11
whereCdoes not depend onx,t.
Note that inequality6.9also true in the casep1. Then by4.11, we get inequality 6.2.
Theorem 6.2. Let 1 ≤ p < ∞and ω1x, tand ω2x, rfulfill condition4.14. Then forp > 1 the singular integral operatorTis bounded from the spaceMp,ω1Rnto the spaceMp,ω2Rnand for p1T is bounded fromM1,ω1RntoWM1,ω2Rn.
Proof. Let 1< p <∞andf∈ Mp,ω1Rn. ByTheorem 6.1we obtain Tf
Mp,ω2 sup
x∈Rn, t>0
ω−12 x, tt−n/pTf
LpBx,t
≤C sup
x∈Rn, t>0
ω−12 x, t ∞
t
r−n/p−1f
LpBx,rdr.
6.12
Hence
Tf
Mp,ω2 ≤Cf
Mp,ω1 sup
x∈Rn, t>0
1 ω2x, t
∞
t
ω1x, rdr r
≤Cf
Mp,ω1
6.13
by4.14, which completes the proof for 1< p <∞.
Letp1 andf∈ M1,ω1Rn. ByTheorem 6.1we obtain Tf
WM1,ω2 sup
x∈Rn, t>0
ω−12 x, tt−nTf
WL1Bx,t
≤C sup
x∈Rn, t>0
ω2−1x, t ∞
t
r−n−1f
L1Bx,rdr.
6.14
Hence
TfWM
1,ω2 ≤CfM
1,ω1Rn sup
x∈Rn, t>0
1 ω2x, t
∞
t
ω1x, rdr r
≤Cf
M1,ω1
6.15
by4.14, which completes the proof forp1.
Remark 6.3. Note that Theorems6.1and6.2were proved in4 see also5.Theorem 6.2 does not impose the pointwise doubling conditions 3.3 and 3.4. In the caseω1x, r ω2x, r ωx, r,Theorem 6.2is containing the results ofTheorem 3.2.
7. The Generalized Morrey Estimates for
the Operators V
γ−Δ V
−βand V
γ∇−Δ V
−βIn this section we consider the Schr ¨odinger operator−Δ V onRn, where the nonnegative potentialV belongs to the reverse H ¨older classB∞Rnfor some q1 ≥ n. The generalized MorreyMp,ωRnestimates for the operatorsVγ−Δ V−βandVγ∇−Δ V−βare obtained.
The investigation of Schr ¨odinger operators on the Euclidean spaceRnwith nonnega- tive potentials which belong to the reverse H ¨older class has attracted attention of a number of authorscf.35–37. Shen36studied the Schr ¨odinger operator−Δ V, assuming the nonnegative potential V belongs to the reverse H ¨older class BqRn for q ≥ n/2 and he proved theLpboundedness of the operators−Δ Viγ,∇2−Δ V−1,∇−Δ V−1/2,and
∇−Δ V−1. Kurata and Sugano generalized Shens results to uniformly elliptic operators in 38. Sugano 39 also extended some results of Shen to the operator Vγ−Δ V−β, 0 ≤ γ ≤ β ≤ 1,andVγ∇−Δ V−β, 0 ≤ γ ≤ 1/2 ≤ β ≤ 1 andβ−γ ≥ 1/2. Later, Lu40 and Li41investigated the Schr ¨odinger operators in a more general setting.
We investigate the generalized MorreyMp,ω1-Mq,ω2boundedness of the operators
T1Vγ−Δ V−β, 0≤γ ≤β≤1, T2Vγ∇−Δ V−β, 0≤γ≤ 1
2 ≤β≤1, β−γ≥ 1 2.
7.1
Note that the operatorsV−Δ V−1andV1/2∇−Δ V−1in41are the special case ofT1
andT2, respectively.
It is worth pointing out that we need to establish pointwise estimates forT1,T2and their adjoint operators by using the estimates of fundamental solution for the Schr ¨odinger operator onRnin41. And we prove the generalized Morrey estimates by usingMp,ω1−Mq,ω2
boundedness of the fractional maximal operators.
LetV ≥0. We sayV ∈B∞, if there exists a constantC >0 such that
V L∞B≤ C
|B|
B
Vxdx 7.2
holds for every ballBinRnsee41.
The following are two pointwise estimates for T1 and T2 which are proven in 37, Lemma 3.2with the potentialV ∈B∞.
Theorem B. SupposeV ∈B∞and 0≤γ≤β≤1. Then there exists a constantC >0 such that T1fx≤CMαfx, f∈C∞0Rn, 7.3
whereα2β−γ.
Theorem C. SupposeV ∈B∞, 0≤γ≤1/2≤β≤1 andβ−γ≥1/2. Then there exists a constant C >0 such that
T2fx≤CMαfx, f∈C∞0Rn, 7.4
whereα2β−γ−1.
The previous theorems will yield the generalized Morrey estimates forT1andT2. Corollary 7.1. Assume thatV ∈B∞, and 0≤γ≤β≤1. Let 1≤p≤q <∞, 2β−γ n1/p−1/q, and condition5.13be satisfied forα 2β−γ. Then forp >1 the operatorT1 is bounded from Mp,ω1RntoMq,ω2Rnand forp1 T1is bounded fromM1,ω1RntoWMq,ω2Rn.
Corollary 7.2. Assume thatV ∈B∞, 0≤γ ≤1/2≤ β≤1,andβ−γ ≥1/2. Let 1≤ p≤q <∞, 2β−γ−1n1/p−1/q,and condition5.13be satisfied forα2β−γ−1. Then forp >1 the operatorT2is bounded fromMp,ω1RntoMq,ω2Rnand forp1 T2is bounded fromM1,ω1Rnto WMq,ω2Rn.
8. Some Applications
The theorems ofSection 2can be applied to various operators which are estimated from above by Riesz potentials. We give some examples.
Suppose thatLis a linear operator onL2which generates an analytic semigroupe−tL with the kernelptx, ysatisfying a Gaussian upper bound, that is,
pt
x, y≤ c1
tn/2e−c2|x−y|2/t 8.1
forx, y∈Rnand allt >0, wherec1, c2>0 are independent ofx,y,andt.
For 0< α < n,the fractional powersL−α/2of the operatorLare defined by
L−α/2fx 1 Γα/2
∞
0
e−tLfx dt
t−α/21. 8.2
Note that ifL−Δis the Laplacian onRn, thenL−α/2is the Riesz potentialIα. See, for example,2, Chapter 5.
Theorem 8.1. Let 0 < α < n, 1 ≤ p < q < ∞,α/n 1/p−1/qand conditions5.13,8.1 are satisfied. Then forp > 1 the operatorL−α/2 is bounded fromMp,ω1Rn toMq,ω2Rn and for p1L−α/2is bounded fromM1,ω1RntoWMq,ω2Rn.
Proof. Since the semigroup e−tL has the kernel ptx, y which satisfies condition 8.1, it follows that
L−α/2fx≤CIαfx 8.3
for allx∈Rn, whereC >0 is independent ofxsee42. Hence byTheorem 5.2we have L−α/2fM
q,ω2 ≤CIαfM
q,ω2 ≤CfM
p,ω1, ifp >1, L−α/2f
WMq,ω2 ≤CIαf
WMq,ω2 ≤Cf
M1,ω1, if p1, 8.4 where the constantC >0 is independent off.
Property8.1is satisfied for large classes of differential operators. We mention two of them.
a Consider a magnetic potential→−a, that is, a real-valued vector potential→−a a1, a2, . . . , an, and an electric potentialV. We assume that for anyk 1,2, . . . , n,ak ∈Lloc2 and 0≤V ∈Lloc1 .The operatorL, which is given by
L−
∇ −i→−a2Vx, 8.5 is called the magnetic Schr ¨odinger operator.
By the well-known diamagnetic inequality see 43, Theorem 2.3 we have the following pointwise estimate. For anyt >0 andf ∈L2,
e−tLf≤e−tΔf, 8.6 which implies that the semigroup e−tL has the kernel ptx, ywhich satisfies upper bound 8.1. bLetA aijx1≤i,j≤n be ann×nmatrix with complex-valued entriesaij ∈ L∞
satisfying
Re
n i,j1
aijxζiζj ≥λ|ζ|2 8.7
for allx∈Rn, ζ ζ1, ζ2, . . . , ζn∈Cnand someλ >0. Consider the divergence form operator Lf≡ −div
A∇f
, 8.8
which is interpreted in the usual weak sense via the appropriate sesquilinear form.
It is known that the Gaussian bound8.1for the kernel ofe−tLholds whenAhas real- valued entriessee, e.g.,44, or whenn1, 2 in the case of complex-valued entriessee45, Chapter 1.
Finally we note that under the appropriate assumptionssee2,46, Chapter 5;45, pages 58-59 one can obtain results similar to Theorem 8.1 for a homogeneous elliptic operatorLinL2of order 2min the divergence form
Lf −1m
|α||β|m
Dα
aαβDβf
. 8.9
In this case estimate8.1should be replaced by pt
x, y≤ c3
tn/2me−c4|x−y|/t1/2m2m/2m−1 8.10 for allt >0 and allx, y∈Rn,wherec3, c4>0 are independent ofx,y,andt.
Acknowledgments
The author thanks the referee for carefuly reading the paper and useful comments. The author was partially supported by the Grant of the Azerbaijan-US Bilateral Grants Program IIProject ANSF Award/AZM1-3110-BA-08.
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