Boundedness of the Maximal,

(1)

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, 2}

1Department of Mathematics, Ahi Evran University, Kirsehir, Turkey

2Institute of Mathematics and Mechanics, Baku, Azerbaijan

Correspondence should be addressed to Vagif S. Guliyev,[email protected] 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 ón-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 ödinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse H ölder 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 ∈L^loc₁ 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,

(2)

Iαfx

Rⁿ

f y

x−ydy^n−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 ∈L^loc₁ 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.

(3)

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 ∈WL^loc_p 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≤υ^α/n−1_n 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.

(4)

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 ∈L^loc_p 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 r^λ−n/p:

Mp,λRⁿ Mp,ωRⁿ|_ωx,rr^λ−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, t^pdt

t ≤C ωx, r^p, 3.4

for the maximal or singular operator and the condition _∞

r

t^αpωx, t^pdt

t ≤C r^αpωx, r^p 3.5

(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,ω

R

ⁿ

Theorem 4.1. Let 1≤p <∞andf∈L^loc_p Rⁿ. Then forp >1 Mf

LpBx,t≤Ct^n/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

χ_Bx,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

(6)

whereCdoes not depend onf. From4.5we have Mf1

2t

r^−n/p−1f

LpBx,rdr

≤Ct^n/p _∞

t

r^−n/p−1f

LpBx,rdr

4.6

easily obtained from the fact that f _L_p_Bx,2tis nondecreasing int, so that f _L_p_Bx,2ton the right-hand side of4.5is dominated by the right-hand side of4.6.

To estimateMf2, we first prove the following auxiliary inequality:

B^x,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:

B^x,t

x−y⁻ⁿf

ydy≤β

B^x,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

B^x,2t∩Bz,r

y−z⁻ⁿf ydy

≤Csup

r≥2t

B^x,2t∩Bz,r

x−y⁻ⁿf ydy

≤C

B^x,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

(7)

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 _L_p_Bx,t. 4.11

Since 1 _L_p_Bx,t Ct^n/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

Mf_M

p,ω2 ≤Cf_M

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 <∞.

(8)

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

Mf_WM

1,ω2 ≤Cf_M

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,ω

R

ⁿ

5.1. Spanne Type Result

Theorem 5.1. Let 1≤p <∞, 0< α < n/p, 1/q1/p−α/n,andf∈L^loc_p Rⁿ. Then forp >1 Iαf

LqBx,t ≤Ct^n/q _∞

t

r^−n/q−1f

LpBx,rdr, 5.1

and forp1

Iαf

WLqBx,t ≤Ct^n/q _∞

t

r^−n/q−1f

L1Bx,rdr, 5.2

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

(9)

Then

Iαf1

LqBx,t≤Cf

LpBx,2t, 5.5 where the constantCis independent off.

Taking into account that f

LpBx,2t≤Ct^n/q _∞

2t

r^−n/q−1f

LpBx,rdr, 5.6

we get

Iαf1

LqBx,t≤Ct^n/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≤

B^x,2t

z−y^α−nfydy

LqBx,t

≤C

B^x,2t

x−y^α−nf

ydyχ_Bx,t

LqRⁿ.

5.8

We chooseβ > n/qand obtain

B^x,2t|x−y|^α−nf

ydyβ

B^x,2t

x−y^α−nβf

y_∞

|^x−y|s^−β−1ds

dy

β _∞

2t

s^−β−1

{y∈Rⁿ:2t≤|x−y|≤s}

x−y^α−nβf ydy

ds

≤C _∞

2t

s^−β−1f

LpBx,s|x−y|^α−nβ

LpBx,sds

≤C _∞

2t

s^α−n/p−1f

LpBx,sds.

5.9

Therefore

Iαf2

LqBx,t ≤Ct^n/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

(10)

By boundedness of the operatorIαfromL1RⁿtoWLqRⁿwe have Iαf1

WL1Bx,t ≤Cf

LqBx,2t, 5.12 whereCdoes not depend onx,t.

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

t^αω1x, 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

r^αω1x, rdr r

5.14

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

r^αω1x, rdr r

≤Cf

M1,ω1

5.16

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.

(11)

5.2. Adams Type Result

Theorem 5.4. Let 1≤p <∞, 0< α < n/p,andf∈L^loc_p Rⁿ. Then Iαfx≤Ct^α Mfx C

_∞

t

r^α−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| ≤ C1t^αMfx.ForIαf2xwe have

Iαf2x≤

B^x,2t

x−y^α−nf ydy

≤C

B^x,2t f

ydy _∞

|x−y|r^α−n−1dr

≤C _∞

2t

2t<|x−y|<r

f ydy

r^α−n−1dr

≤C _∞

t

r^α−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

t^αωx, t _∞

t

r^α ωx, rdr

r ≤Cωx, t^p/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,ωRⁿtoM_q,ω^p/qRⁿand forp1 the operatorsMαandIαare bounded fromM1,ωRⁿtoWM_q,ω^1/qRⁿ.

Proof. In view of the well-known pointwise estimateMαfx≤CIα|f|x, it suﬃces to treat only the case of the operatorIα.

(12)

Let 1≤p <∞andf∈ Mp,ωRⁿ. ByTheorem 5.4we get Iαfx≤Cr^α Mfx Cf

Mp,ω

_∞

r

t^αωx, tdt

t . 5.22

From5.20we haver^αωx, r ≤Cωx, r^p/q. Making also use of condition5.20, we obtain Iαfx≤Cωx, r^p/q−1 Mfx Cωx, r^p/q f

Mp,ω. 5.23

Sinceωx, ris surjective, we can chooser >0 so thatωx, r Mfx f ⁻¹_M_p,ω_Rn, assuming thatfis not identical 0. Hence, for everyx∈Rⁿ, we have

Iαfx≤C

Mfxp/q f^1−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

Iαf_M

q,ωp/q sup

x∈Rⁿ, t>0

ωx, t^−p/qt^−n/qIαf

LqBx,t

≤Cf^1−p/q

Mp,ω sup

x∈Rⁿ, t>0

ωx, t^−p/qt^−n/qMf^p/q

LpBx,t

≤Cf

Mp,ω,

5.25

if 1< p < q <∞and Iαf

WM_q,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/qMf^1/q

WL1Bx,t

≤Cf_M

1,ω,

5.26

ifp1< q <∞.

6. Singular Operators in the Spaces M

p,ω

R

ⁿ

Theorem 6.1. Let 1≤p <∞andf∈L^loc_p Rⁿ. Then forp >1 Tf

t

r^−n/p−1f

LpBx,rdr, 6.1

(13)

and forp1

Tf

WL1Bx,t ≤Ctⁿ _∞

t

r⁻ⁿ⁻¹f

L1Bx,rdr, 6.2

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

2t

r^−n/p−1f

LpBx,rdr, 6.5

we get

Tf1

2t

r^−n/p−1f

LpBx,rdr. 6.6

To estimate Tf2 LpBx,t, we observe that Tf2z≤C

B^x,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

2t

r^−n/p−1f

LpBx,rdr. 6.9

From6.6and6.9we arrive at6.1.

(14)

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.

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

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

(15)

Hence

Tf_WM

1,ω2 ≤Cf_M

1,ω1Rⁿ sup

x∈Rⁿ, t>0

1 ω2x, t

_∞

t

ω1x, rdr r

≤Cf

M1,ω1

6.15

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

^−β

and V

^γ

∇−Δ V

^−β

In 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 operatorsV^γ−Δ V^−βandV^γ∇−Δ V^−βare obtained.

The investigation of Schr ödinger operators on the Euclidean spaceRⁿwith nonnegative potentials which belong to the reverse H ölder class has attracted attention of a number of authorscf.35–37. Shen36studied the Schr ödinger operator−Δ V, assuming the nonnegative potential V belongs to the reverse H ölder class BqRⁿ for q ≥ n/2 and he proved theLpboundedness of the operators−Δ Vîγ,∇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 V^γ−Δ V^−β, 0 ≤ γ ≤ β ≤ 1,andV^γ∇−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

T1V^γ−Δ V^−β, 0≤γ ≤β≤1, T2V^γ∇−Δ V^−β, 0≤γ≤ 1

2 ≤β≤1, β−γ≥ 1 2.

7.1

Note that the operatorsV−Δ V⁻¹andV^1/2∇−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.

(16)

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

t^n/2e^−c²|^x−y|²^/t 8.1

forx, y∈Rⁿand allt >0, wherec1, c2>0 are independent ofx,y,andt.

(17)

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/2f_M

q,ω2 ≤CIαf_M

q,ω2 ≤Cf_M

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 ∈L^loc₂ and 0≤V ∈L^loc₁ .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 aijx_1≤i,j≤n be ann×nmatrix with complex-valued entriesaij ∈ L∞

satisfying

Re

n i,j1

aijxζiζj ≥λ|ζ|² 8.7

(18)

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 −1^m

|α||β|m

D^α

aαβD^βf

. 8.9

In this case estimate8.1should be replaced by pt

x, y≤ c3

t^n/2me^−c⁴^|x−y|/t^1/2m^2m/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.

References

1 R. R. Coifman and Y. Meyer, Au Delà des Opérateurs Pseudo-Différentiels, vol. 57 of Astérisque, Société Mathématique de France, Paris, France, 1978.

2 E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.

3 E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, no. 3, Princeton University Press, Princeton, NJ, USA, 1971.

4 V. S. Guliyev, Integral Operators on Function Spaces on the Homogeneous Groups and on Domains inRⁿ, Doctor’s dissertation, Steklov Mathematical Institute, Moscow, Russia, 1994.

5 V. S. Guliyev, “Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications,” Baku, pp. 1–332, 1999Russian.

6 V. I. Burenkov and H. V. Guliyev, “Necessary and suﬃcient conditions for boundedness of the maximal operator in local Morrey-type spaces,” Studia Mathematica, vol. 163, no. 2, pp. 157–176, 2004.

7 V. I. Burenkov and V. S. Guliyev, “Necessary and suﬃcient conditions for the boundedness of the Riesz potential in local morrey-type spaces,” Potential Analysis, vol. 31, no. 2, pp. 1–39, 2009.

8 V. I. Burenkov, V. S. Guliev, and G. V. Guliev, “Necessary and suﬃcient conditions for the boundedness of the Riesz potential in local Morrey-type spaces,” Doklady Mathematics, vol. 75, no. 1, pp. 103–107, 2007, Translated from Doklady Akademii Nauk, vol. 412, no. 5, pp. 585–589, 2007.

9 V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev, “Necessary and suﬃcient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces,” Journal of Computational and Applied Mathematics, vol. 208, no. 1, pp. 280–301, 2007.