Potential Operators in Variable Exponent Lebesgue Spaces: Two Weight Estimates

Journal of Inequalities and Applications Volume 2010, Article ID 329571,27pages doi:10.1155/2010/329571

Research Article

Potential Operators in Variable Exponent Lebesgue

Spaces: Two-Weight Estimates

Vakhtang Kokilashvili,

_{Alexander Meskhi,}

and Muhammad Sarwar

1_{Department of Mathematical Analysis, A. Razmadze Mathematical Institute, 1. M. Aleksidze Street,}

0193 Tbilisi, Georgia

2_{Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, 2 University Street,}

0143 Tbilisi, Georgia

3_{Department of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University,}

77 Kostava Street, 0175 Tbilisi, Georgia

4_{Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town,}

Lahore 54600, Pakistan

Correspondence should be addressed to Alexander Meskhi,[email protected]

Received 17 June 2010; Accepted 24 November 2010

Academic Editor: M. Vuorinen

Copyrightq2010 Vakhtang Kokilashvili 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.

Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces X, d, μ are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for corresponding inequalities. Appropriate examples of weights are also given.

1. Introduction

We study the two-weight problem for Hardy-type and potential operators in Lebesgue spaces with nonstandard growth defined on quasimetric measure spacesX, d, μ. In particular, our aim is to derive easily verifiable sufficient conditions for the boundedness of the operators

Tα·f

x

X

fy

μBx, dx, y1−αxdμ

y, Iα·f

x

X

fy

dx, y1−αxdμ

y

1.1

inequalities when the weights are of special type and the exponentpof the space is constant. We assume that the exponent p satisfies the local log-H ¨older continuity condition, and if the diameter ofX is infinite, then we suppose that pis constant outside some ball. In the framework of variable exponent analysis such a condition first appeared in the paper 1, where the author established the boundedness of the Hardy-Littlewood maximal operator in

Lp·_Rn_{. As far as we know, unfortunately, an analog of the log-H ¨older decay conditionat} infinityforp:X → 1,∞is not known even in the unweighted case, which is well-known and natural for the Euclidean spacessee2–5. Local log-H ¨older continuity condition for the exponentp, together with the log-H ¨older decay condition, guarantees the boundedness of operators of harmonic analysis inLp·_Rn_{spacessee, e.g.,6. The technique developed} here enables us to expect that results similar to those of this paper can be obtained also for other integral operators, for instance, for maximal and Calder ´on-Zygmund singular operators defined onX.

Considerable interest of researchers is focused on the study of mapping properties of integral operators defined onquasimetric measure spaces. Such spaces with doubling measure and all their generalities naturally arise when studying boundary value problems for partial differential equations with variable coefficients, for instance, when the quasimetric might be induced by a differential operator or tailored to fit kernels of integral operators. The problem of the boundedness of integral operators naturally arises also in the Lebesgue spaces with nonstandard growth. Historically the boundedness of the maximal and fractional integral operators inLp·_{Xspaces was derived in the papers7–14. Weighted inequalities} for classical operators inLpw·spaces, wherewis a power-type weight, were established in the papers10–12,15–19, while the same problems with general weights for Hardy, maximal, and fractional integral operators were studied in10,20–25. Moreover, in the latter paper, a complete solution of the one-weight problem for maximal functions defined on Euclidean spaces is given in terms of Muckenhoupt-type conditions.

It should be emphasized that in the classical Lebesgue spaces the two-weight problem for fractional integrals is already solved see 26, 27, but it is often useful to construct concrete examples of weights from transparent and easily verifiable conditions.

To derive two-weight estimates for potential operators, we use the appropriate inequalities for Hardy-type transforms on X which are also derived in this paper and Hardy-Littlewood-Sobolev-type inequalities forTα·andIα·inLp·Xspaces.

The paper is organized as follows: inSection 1, we give some definitions and prove auxiliary results regarding quasimetric measure spaces and the variable exponent Lebesgue spaces;Section 2is devoted to the sufficient governing two-weight inequalities for Hardy-type operators defined on quasimetric measure spaces, while inSection 3we study the two-weight problem for potentials defined onX.

Finally we point out that constants often different constants in the same series of inequalities will generally be denoted by c or C. The symbol fx ≈ gx means that there are positive constants c1 and c2 independent of x such that the inequality fx ≤ c1gx≤c2fxholds. Throughout the paper is denoted the functionpx/px−1by the

symbolpx.

2. Preliminaries

real-valued functionquasimetricdonX×Xsatisfying the conditions:

idx, y 0 if and only ifxy;

iithere exists a constanta1>0, such thatdx, y≤a1dx, z dz, yfor allx, y, z∈ X;

iiithere exists a constanta0>0, such thatdx, y≤a0dy, xfor allx, y,∈X.

We assume that the ballsBx, r : {y ∈ X : dx, y < r}are measurable and 0 ≤

μBx, r<∞for allx∈Xandr >0; for every neighborhoodVofx∈X, there existsr >0, such thatBx, r⊂V. Throughout the paper we also suppose thatμ{x}0 and that

Bx, R\Bx, r/∅, 2.1

for allx∈X, positiverandRwith 0< r < R < L, where

L:diamX supdx, y:x, y∈X. 2.2

We call the triple X, d, μ a quasimetric measure space. If μ satisfies the doubling conditionμBx,2r≤cμBx, r, where the positive constantcdoes not depend onx∈X

and r > 0, thenX, d, μis called a space of homogeneous typeSHT. For the definition, examples, and some properties of an SHT see, for example, monographs28–30.

A quasimetric measure space, where the doubling condition is not assumed, is called a nonhomogeneous space.

Notice that the conditionL <∞implies thatμX<∞because we assumed that every ball inXhas a finite measure.

We say that the measureμis upper AhlforsQ-regular if there is a positive constantc1

such thatμBx, r ≤ c1rQ for for allx ∈X andr > 0. Further,μis lower AhlforsQ-regular

if there is a positive constantc2 such thatμBx, r ≥ c2rq for allx ∈ X andr > 0. It is easy

to check that ifX, d, μis a quasimetric measure space andL <∞, thenμis lower Ahlfors regularsee also, e.g.,8for the case whendis a metric.

For the boundedness of potential operators in weighted Lebesgue spaces with constant exponents on nonhomogeneous spaces we refer, for example, to the monograph31, Chapter 6and references cited therein.

Letpbe a nonnegativeμ-measurable function onX. Suppose thatEis aμ-measurable set inX. We use the following notation:

p−E:inf

E p; p E:sup_E p; p−:p−X; p :p X;

Bx, r:y∈X :dx, y≤r, kBx, r:Bx, kr; Bxy:B

x, dx, y;

Bxy :B

x, dx, y; gB: 1

μB

B

gxdμx.

Assume that 1 ≤ p− ≤ p < ∞. The variable exponent Lebesgue space Lp·X

sometimes it is denoted byLpx_{Xis the class of allμ-measurable functionsf onX for} whichSpf:

X|fx|

px_{dμx<∞. The norm inL}p·_{Xis defined as follows:}

f _Lp·_Xinf

λ >0 :Sp

_f

λ

≤1

. 2.4

It is knownsee, e.g.,8,15,32,33thatLp·_{is a Banach space. For other properties of} Lp·_{spaces we refer, for example, to32–34.}

We need some definitions for the exponentpwhich will be useful to derive the main results of the paper.

Definition 2.1. Let X, d, μ be a quasimetric measure space and let N ≥ 1 be a constant. Suppose thatpsatisfies the condition 0 < p− ≤ p < ∞. We say thatp belongs to the class

PN, x, wherex∈X, if there are positive constantsbandcwhich might be depended on

xsuch that

μBx, Nrp−Bx,r−pBx,r_{≤c 2.5}

holds for allr, 0< r ≤ b. Further,p ∈ PNif there are positive constantsbandcsuch that

2.5holds for allx∈Xand allrsatisfying the condition 0< r≤b.

Definition 2.2. LetX, d, μbe an SHT. Suppose that 0< p−≤p <∞. We say thatp∈LHX, x

psatisfies the log-H ¨older-type condition at a pointx ∈Xif there are positive constantsb

andcwhich might be depended onxsuch that

_px−p

y≤ c

−lnμBxy

2.6

holds for allysatisfying the condition dx, y ≤ b. Further,p ∈ LHX psatisfies the log-H ¨older type condition onXif there are positive constantsbandcsuch that2.6holds for allx, ywithdx, y≤b.

We will also need another form of the log-H ¨older continuity condition given by the following definition.

Definition 2.3. LetX, d, μbe a quasimetric measure space, and let 0< p₋ ≤p <∞. We say

thatp ∈ LHX, xif there are positive constantsbandc which might be depended onx

such that

_px−p

y≤ c

−lndx, y 2.7

It is easy to see that if a measureμis upper AhlforsQ-regular andp∈LHX resp.,

p∈LHX, x, thenp∈LHX resp.,p∈LHX, x. Further, ifμis lower AhlforsQ-regular andp∈LHX resp.,p∈LHX, x, thenp∈LHX resp.,p∈LHX, x.

Remark 2.4. It can be checked easily that ifX, d, μis an SHT, thenμBx0x ≈μBxx0.

Remark 2.5. LetX, d, μbe an SHT with L < ∞. It is knownsee, e.g., 8, 35 that ifp ∈

LHX, thenp∈ P1. Further, ifμis upper AhlforsQ-regular, then the conditionp ∈ P1 implies thatp∈LHX.

Proposition 2.6. Letcbe positive and let1 < p−X ≤ p X < ∞andp ∈ LHX(resp.,p ∈

LHX, then the functions cp·, 1/p·, andp· belong toLHX resp., LHX. Further if

p ∈ LHX, x resp.,p ∈ LHX, xthen cp·, 1/p·, andp·belong toLHX, x resp.,p ∈

LHX, x.

The proof of the latter statement can be checked immediately using the definitions of the classes LHX, x, LHX, LHX, x, and LHX.

Proposition 2.7. LetX, d, μbe an SHT and letp∈ P1. ThenμBxypx ≤cμByxpyfor all

x, y∈XwithμBx, dx, y≤b, wherebis a small constant, and the constantcdoes not depend onx, y∈X.

Proof. Due to the doubling condition for μ, Remark 1.1, the condition p ∈ P1 and the fact that x ∈ By, a1a0 1dy, x we have the following estimates: μBxypx ≤

μBy, a1a0 1dx, ypx ≤cμBy, a1a0 1dx, ypy ≤ cμByxpy, which proves the statement.

The proof of the next statement is trivial and follows directly from the definition of the classesPN, xandPN. Details are omitted.

Proposition 2.8. LetX, d, μbe a quasimetric measure space and letx0 ∈X. Suppose thatN ≥1

be a constant. Then the following statements hold:

iifp ∈ PN, x0(resp.,p ∈ PN, then there are positive constantsr0,c1, andc2 such

that for all0< r ≤r0and ally∈Bx0, r(resp., for allx0, ywithdx0, y< r≤r0), one

has thatμBx0, Nrpx0≤c1μBx0, Nrpy ≤c2μBx0, Nrpx0.

iiLetp ∈ PN, x0, then there are positive constantsr0,c1, andc2 (in general, depending

on x0) such that for allr (r ≤ r0) and allx, y ∈ Bx0, rone has μBx0, Nrpx ≤ c1μBx0, Nrpy≤c2μBx0, Nrpx.

iiiLetp∈ PN, then there are positive constantsr0,c1, andc2such that for all ballsBwith

radiusr(r≤r0) and allx, y∈B, one has thatμNBpx≤c1μNBpy≤c2μNBpx.

It is known thatsee, e.g., 32, 33 if f is a measurable function on X and E is a measurable subset ofX, then the following inequalities hold:

f _Lpp·E_E≤Sp

fχE

≤ f p_L−p·E_E, f _Lp·_E≤1;

_f p−E

Lp·_E≤Sp

fχE

≤ f p_Lp·E_E, f Lp·_E >1.

Further, H ¨older’s inequality in the variable exponent Lebesgue spaces has the following form:

E

fgdμ≤

1

p−E

1

p₋E

f _Lp·_E g _Lp·_E. 2.9

Lemma 2.9. LetX, d, μbe an SHT.

iIfβis a measurable function onXsuch thatβ < −1and ifr is a small positive number, then there exists a positive constantcindependent ofrandxsuch that

X\Bx0,r

μBx0y

βx_dμy≤cβx 1

βx μBx0, r

βx 1_{. 2.10}

iiSuppose thatpandαare measurable functions onX satisfying the conditions1 < p₋ ≤ p <∞andα−>1/p−. Then there exists a positive constantcsuch that for allx∈Xthe

inequality

Bx0,2dx0,x

μBx, dx, yαx−1pxdμy≤cμBx0, dx0, x

αx−1px 1

2.11

holds.

Proof. Partiwas proved in35 see also31, page 372, for constantβ. The proof of Partii is given in31,Lemma 6.5.2, page 348for constantαandp, but repeating those arguments we can see that it is also true for variableαandp. Details are omitted.

Lemma 2.10. LetX, d, μbe an SHT. Suppose that0< p₋ ≤p <∞, thenpsatisfies the condition

p∈ P1(resp.,p∈ P1, x) if and only ifp∈LHX resp.,p∈LHX, x.

Proof. We follow1.

Necessity. Let p ∈ P1, and letx, y ∈ X with dx, y < c0 for some positive constantc0.

Observe thatx, y ∈B, whereB : Bx,2dx, y. By the doubling condition forμ, we have thatμBxy−|px−py| ≤cμB−|px−py|≤ cμBp−B−pB ≤ C, whereCis a positive constant which is greater than 1. Taking now the logarithm in the last inequality, we have thatp ∈

LHX. Ifp∈ P1, x, then by the same arguments we find thatp∈LHX, x.

Sufficiency. Let B : Bx0, r. First observe that If x, y ∈ B, then μBxy ≤ cμBx0, r.

Consequently, this inequality and the condition p ∈ LHX yield |p−B − p B| ≤

C/ − lnc0μBx0, r. Further, there exists r0 such that 0 < r0 < 1/2 and c1 ≤

lnμB/−lnc0μB ≤ c2, 0 < r ≤ r0, where c1 and c2 are positive constants. Hence

μBp−B−pB_≤μBC/lnc0μB _{expClnμB/lnc}

Let, now,p∈LHX, xand letBx :Bx, rwherer is a small number. We have that

p Bx−px≤c/−lnc0μBx, randpx−p−Bx≤c/−lnc0μBx, rfor some positive

constantc0. Consequently,

μBx

p−Bx−pBx_μB x

px−pBx_μB x

p−Bx−px_≤cμB x

₋2c/−lnc0μBx_≤C.

2.12

Definition 2.11. A measure μ on X is said to satisfy the reverse doubling condition μ ∈

RDCX if there exist constantsA > 1 and B > 1 such that the inequality μBa, Ar ≥ BμBa, rholds.

Remark 2.12. It is known that if all annulus inX are not emptyi.e., condition2.1holds, thenμ∈ DCXimplies thatμ∈ RDCX see, e.g.,28, page 11, Lemma 20.

Lemma 2.13. Let X, d, μ be an SHT. Suppose that there is a point x0 ∈ X such that p ∈

LHX, x0. LetAbe the constant defined inDefinition 2.11. Then there exist positive constantsr0

andC(which might be depended onx0) such that for allr,0< r ≤r0, the inequality

μBA

p−BA−pBA_{≤C 2.13}

holds, whereBA:Bx0, Ar\Bx0, rand the constantCis independent ofr.

Proof. Taking into account condition 2.1 and Remark 2.12, we have that μ ∈ RDCX. Let B : Bx0, r. By the doubling and reverse doubling conditions, we have thatμBA

μBx0, Ar−μBx0, r ≥ B −1μBx0, r ≥ cμAB. Suppose that 0 < r < c0, where c0

is a sufficiently small constant. Then by usingLemma 2.10we find thatμBAp−BA−pBA ≤

cμABp−BA−pBA_≤cμABp−AB−pAB_≤c.

In the sequel we will use the notation:

I1,k :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

B

x0,A

k−1_L a1

ifL <∞,

B

x0, Ak−1

a1

ifL∞,

I2,k :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Bx0,Ak 2a1L

\B

x0, Ak−1_L

a1

if L <∞,

Bx0, Ak 2a1

\B

x0, Ak−1

a1

if L∞,

I3,k :

⎧ ⎨ ⎩

X\Bx0, Ak 2La1

if L <∞,

X\Bx0, Ak 2a1

Ek:

⎧ ⎨ ⎩

Bx0, Ak 1L

\Bx0, AkL

if L <∞,

Bx0, Ak 1

\Bx0, Ak

if L∞,

2.14

where the constantsAand a1 are taken, respectively, from Definition 2.11and the triangle

inequality for the quasimetricd, andLis a diameter ofX.

Lemma 2.14. LetX, d, μbe an SHT and let1 < p−x ≤px ≤qx≤q X< ∞. Suppose

that there is a pointx0∈Xsuch thatp, q∈LHX, x0. Assume that ifL∞, thenpx≡pc≡const andqx≡qc≡const outside some ball Bx0,a. Then there exists a positive constant C such that

k

fχI2,k Lp·_X gχI2,k Lq·_X≤C f _Lp·_X g _Lq·_X, 2.15

for allf∈Lp·_Xandg∈Lq·_X.

Proof. Suppose thatL ∞. To prove the lemma, first observe thatμEk ≈ μBx0, Akand μI2,k ≈ μBx0, Ak−1. This holds because μ satisfies the reverse doubling condition and,

consequently,

μEkμ

Bx0, Ak 1

\Bx0, Ak

μBx0, Ak 1

−μBx0, Ak

μBx0, AAk

−μBx0, Ak

≥BμBx0, Ak

−μBx0, Ak

B−1μBx0, Ak

.

2.16

Moreover, the doubling condition yieldsμEk ≤ μBx0, AAk ≤ cμBx0, Ak, where c > 1.

Hence,μEk≈μBx0, Ak.

Further, since we can assume thata1≥1, we find that

μI2,kμ

Bx0, Ak 2a1

\B

x0, Ak−1

a1

μBx0, Ak 2a1

−μB

x0, Ak−1

a1

μBx0, AAk 1a1

−μB

x0, Ak−1

a1

≥BμBx0, Ak 1a1

−μB

x0, Ak−1

a1

≥B2μB

x0, Ak

a1

−μB

x0, Ak−1

a1

≥B3μB

x0, Ak−1

a1

−μB

x0, Ak−1

a1

B3−1μB

x0, Ak−1

a1

.

2.17

Moreover, using the doubling condition for μ we have that μI2,k ≤ μBx0, Ak 2r ≤ cμBx0, Ak 1r ≤ c2μBx0, Ak/a1 ≤ c3μBx0, Ak−1/a1. This gives the estimates B3 −

For simplicity, assume thata1. Suppose thatm0is an integer such thatAm0−1/a1>1.

Let us split the sum as follows:

i

_fχI

2,i Lp·_X· gχI2,i Lq·_X

i≤m0

· · ·

i>m0

· · · :J1 J2. 2.18

Sincepx≡pcconst, qx qcconst outside the ballBx0,1, by using H ¨older’s

inequality and the fact thatpc≤qc, we have

J2

i>m0

fχI2,i _Lpc_X· gχI2,i _Lqc_X≤c f _Lp·_X· g _Lq·_X. 2.19

Let us estimate J1. Suppose that f_Lp·_X ≤ 1 and g_Lq·_X ≤ 1. Also, by

Proposition 2.6, we have that 1/q ∈ LHX, x0. Therefore, by Lemma 2.13 and the fact

that 1/q ∈ LHX, x0, we obtain that μI2,k1/qI2,k ≈ χI2,kLq·_X ≈ μI2,k1/q−I2,k and

μI2,k1/q

_I

2,k ≈ _χ

I2,k_Lq·_X ≈ μI2,k1/q

−Ik_{, where k} ≤ _{m0. Further, observe that these}

estimates and H ¨older’s inequality yield the following chain of inequalities:

J1≤c

k≤m0

Bx0,Am01

fχI2,k Lp·_X· gχI2,k Lq·_X

χI2,k Lq·_X· χI2,k Lq·_X

χEkxdμx

c

Bx0,Am01

k≤m0

fχI2,k _Lp·_X· gχI2,k _Lq·_X

_χI

2,k _Lq·_X· χI2,k _Lq·_X

χEkxdμx

≤c

k≤m0 _fχI

2,k Lp·_X

_χI

2,k Lq·_X

χEkx

Lq·_Bx0,Am01

× k≤m0

gχI2,k _Lq·_X

χI2,k _Lq·_X

χEkx

Lq·_Bx

0,Am01

:cS1

f·S2

g.

2.20

Now we claim thatS1f≤cIf, where

If: k≤m0

_fχI

2,k _Lp·_X

_χI

2k Lp·_X

χEk·

Lp·_Bx0,Am01

, 2.21

and the positive constantcdoes not depend onf. Indeed, suppose thatIf≤1. Then taking into accountLemma 2.13we have that

k≤m0

1

μI2,k

Ek

_fχI 2,k

px

Lp·_Xdμx

≤c

Bx0,Am01

k≤m0 _fχI

2,k Lp·_X

χI2,k Lp·_X

χEkx

px

dμx≤c.

Consequently, sincepx≤qx, Ek⊆I2,kandf_Lp·_X≤1, we find that

k≤m0

1

μI2,k

Ek

_fχI 2,k

qx

Lp·_Xdμx≤

k≤m0

1

μI2,k

Ek

_fχI 2,k

px

Lp·_Xdμx≤c. 2.23

This implies thatS1f≤c. Thus, the desired inequality is proved. Further, let us introduce

the following function:

Py:

k≤2

p I2,kχEky. 2.24

It is clear thatpy≤PybecauseEk⊂I2,k. Hence

If≤c

k≤m0

fχI2,k Lp·_X

χI2k _Lp·_X

χEk·

LP·_Bx0,Am01

2.25

for some positive constantc. Then, by using this inequality, the definition of the functionP, the conditionp∈LHX, and the obvious estimateχI2,k

pI2,k

Lp·_X≥cμI2,k, we find that

Bx0,Am01

k≤m0

fχI2,k _Lp·_X

χI2,k _Lp·_X

χEkx

Px dμx

Bx0,Am01 ⎛

⎝

k≤m0 _fχI

2,k pI2,k

Lp·_X

_χI 2,k

pI2,k

Lp·_X

χEkx

⎞ ⎠_dμx

≤c

Bx0,Am01 ⎛

⎝

k≤m0

fχI2,k pI2,k

Lp·_X

μI2,k

χEkx

⎞

⎠_dμx≤c

k≤m0 _fχI

2,k pI2,k

Lp·_X

≤c

k≤m0

I2,k

fxpxdμx≤c

X

fxpxdμx≤c.

2.26

Consequently,If≤ cf_Lp·_X. Hence,S1f≤ cf_Lp·_X. Analogously taking into

account the fact thatq∈DLXand arguing as above, we find thatS2g≤cg_Lq·_X. Thus,

summarizing these estimates we conclude that

i≤m0

fχIi Lp·_X gχIi Lq·_X ≤c f _Lp·_X g _Lq·_X. 2.27

3. Hardy-Type Transforms

In this section, we derive two-weight estimates for the operators:

Tv,wfx vx

Bx0x

fywydμy, Tv,wfx vx

X\Bx0x

fywydμy.

3.1

Letabe a positive constant, and letpbe a measurable function defined onX. Let us introduce the notation:

p0x:p−

Bx0x

; p0x:

⎧ ⎨ ⎩

p0x ifdx0, x≤a;

pcconst ifdx0, x> a.

p1x:p−

Bx0, a\Bx0x

; p1x:

⎧ ⎨ ⎩

p1x if dx0, x≤a;

pcconst ifdx0, x> a.

3.2

Remark 3.1. If we deal with a quasimetric measure space withL < ∞, then we will assume

thataL. Obviously,p0≡p0andp1 ≡p1in this case.

Theorem 3.2. LetX, d, μbe a quasimetric measure space. Assume that p and qare measurable functions onXsatisfying the condition1 < p− ≤p0x≤qx≤q <∞. In the case whenL∞,

suppose thatp≡pc≡const,q≡qc≡const, outside some ballBx0, a. If the condition

A1: sup 0≤t≤L

t<dx0,x≤L

vxqx

dx0,x≤t

wp0x_ydμy

qx/p0x

dμx<∞, 3.3

holds, thenTv,wis bounded fromLp·XtoLq·X.

Proof. Here we use the arguments of the proofs of Theorem 1.1.4 in31,see page 7and of Theorem 2.1 in21. First, we notice thatp₋ ≤ p0x ≤pxfor allx∈ X. Letf ≥0 and let Spf≤1. First, assume thatL <∞. We denote

Is:

dx0,y<s

fywydμy fors∈0, L. 3.4

Suppose thatIL < ∞, thenIL∈ 2m_,2m 1_{for somem∈ Z. Let us denotes}

j : sup{s :

Is≤2j_{}, j ≤m, ands}

thatIsj ≤2j, Is>2jfors > sj, and 2j ≤

sj≤dx0,y≤sj1fywydμy. Ifβ:limj→ −∞sj,

thendx0, x < Lif and only ifdx0, x ∈ 0, β∪

m

j−∞sj, sj 1. If IL ∞, then we take m ∞. Since 0 ≤ Iβ ≤ Isj ≤ 2j for every j, we have that Iβ 0. It is obvious that

X_j≤m{x:sj< dx0, x≤sj 1}. Further, we have that

Sq

Tv,wf

X

Tv,wfx

qx_dμx

X

vx

Bx0,dx0,x

fywydμy

qx

dμx

X

vxqx

Bx0,dx0,x

fywydμy

qx dμx

≤ m j−∞

sj<dx0,x≤sj1

vxqx

dx0,y<sj1

fywydμy

qx dμx.

3.5

Let us denote

Bjx0:

x∈X :sj−1≤dx0, x≤sj

. 3.6

Notice that Isj 1 ≤ 2j 1 ≤ 4

Bjx0 wyfydμy. Consequently, by this estimate and

H ¨older’s inequality with respect to the exponentp0xwe find that

Sq

Tv,wf

≤c

m

j−∞

sj<dx0,x≤sj1

vxqx

Bjx0

fywydμy

qx dμx

≤c

m

j−∞

sj<dx0,x≤sj1

vxqxJkxdμx,

3.7

where

Jkx:

Bjx0

fyp0x_dμy

qx/p0x

Bjx0

wyp0x_dμy

qx/p0x

Observe now thatqx≥p0x. Hence, this fact and the conditionSpf≤1 imply that

Jkx≤c

Bjx0∩{y:fy≤1}

fyp0x_dμy

Bjx0∩{y:fy>1}

fypydμy

qx/p0x

×

Bjx0

wyp0x_dμyqx/p0 _x

≤c

μBjx0

Bjx0∩{y:fy>1}

fypydμy

×

Bjx0

wyp0x_dμy

qx/p0x .

3.9

It follows now that

Sq

Tv,wf

≤c

⎛

⎝m

j−∞

μBjx0

sj<dx0,x≤sj1

vxqx

×

Bjx0

wyp0x_dμy

qx/p0x dμx

m

j−∞

Bjx0∩{y:fy>1}

fypydμy

sj<dx0,x≤sj1

vxqx

×

Bjx0

wyp0x_dμy

qx/p0x dμx

⎞

⎠_{:cN1 N2.}

3.10

SinceL <∞, it is obvious that

N1≤A1

m1

j−∞

μBjx0

≤CA1,

N2≤A1

m1

j−∞

Bjx0

fypydμy≤C

X

fypydμyA1Sp

f≤A1.

3.11

Let us now suppose thatL∞. We have

Tv,wfx χBx0,axvx

Bx0x

fywydμy

χX\Bx0,axvx

Bx0x

fywydμy:Tv,w1fx Tv,w2fx.

3.12

By using the already proved result forL <∞and the fact that diamBx0, a<∞, we

find thatTv,w1fLq·_Bx

0,a≤cfLp·Bx0,a≤cbecause

A₁a : sup

0≤t≤a

t<dx0,x≤a

vxqx

dx0,x≤t

wp0x_ydμy

qx/p0x

dμx≤A1<∞.

3.13

Further, observe that

Tv,w2fx χX\Bx0,axvx

Bx₀x

fywydμyχX\Bx0,axvx

×

dx0,y≤a

fywydμy

χX\Bx0,axvx

a≤dx0,y≤dx0,x

fywydμy:Tv,w2,1fx Tv,w2,2fx.

3.14

It is easy to see thatsee also31, Theorems 1.1.3 or 1.1.4the condition

A₁a:sup t≥a

dx0,x≥t

vxqc_dμx

1/qc

a≤dx0,y≤t

wypcdμy

1/pc

<∞ 3.15

guarantees the boundedness of the operator

Tv,wfx vx

a≤dx0,y<dx0,x

fromLpc_X\_Bx

0, atoLqcX\Bx0, a. Thus,Tv,w2,2 is bounded. It remains to prove that

Tv,w2,1is bounded. We have

_T2,1 v,w f

Lp·_X

Bx0,ac

vxqc_dμx

1/qc

Bx0,a

fywydμy

≤

Bx0,ac

vxqc_dμx

1/qc

f _Lp·_Bx

0,awLp·Bx0,a.

3.17

Observe, now, that the conditionA1<∞guarantees that the integral

Bx0,ac

vxqc_{dμx 3.18}

is finite. Moreover,N:w_Lp·_Bx

0,a<∞. Indeed, we have that

N≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Bx0,a

wypydμy

1/p−Bx0,a

ifw_Lp·_Bx0,a≤1,

Bx0,a

wypydμy

1/pBx0,a

ifw_Lp·_Bx

0,a>1.

3.19

Further,

Bx0,a

wypydμy

Bx0,a∩{w≤1}

wypydμy

Bx0,a∩{w>1}

wypydμy:I1 I2.

3.20

ForI1, we have thatI1≤μBx0, a<∞. SinceL∞and condition2.1holds, there exists

a pointy0 ∈ X such thata < dx0, y0 < 2a. Consequently,Bx0, a ⊂ Bx0, dx0, y0and py≥p−Bx0, dx0, y0 p0y0, wherey∈Bx0, a. Consequently, the conditionA1<∞

yieldsI2 ≤

Bx0,awy

p0y0_{dy <} ∞. Finally, we have that_T2,1

v,wfLp·_X ≤C. Hence,Tv,w is bounded fromLp·_XtoLq·_X.

Theorem 3.3. LetX, d, μbe a quasimetric measure space. Assume that p and qare measurable functions onX satisfying the condition1 < p₋ ≤ p1x ≤ qx ≤ q < ∞. IfL ∞, then, one

assumes thatp≡pc≡const,q≡qc≡const outside some ballBx0, a. If

B1 sup 0≤t≤L

dx0,x≤t

vxqx

t≤dx0,x≤L

wp1x_ydμy

qx/p1x

dμx<∞, 3.21

thenT_v,w is bounded fromLp·_XtoLq·_X.

Remark 3.4. If p ≡ const, then the condition A1 < ∞ in Theorem 3.2 resp., B1 < ∞ in

Theorem 3.3 is also necessary for the boundedness of Tv,w resp., Tv,w from Lp·X to

Lq·_{X. See31, pages 4-5for the details.}

4. Potentials

In this section, we discuss two-weight estimates for the potential operatorsTα· andIα· on quasimetric measure spaces, where 0< α₋ ≤ α < 1. Ifα ≡const , then we denoteTα·and Iα·byTαandIα, respectively.

The boundedness of Riesz potential operators inLp·_{Ωspaces, whereΩis a domain} inRnwas established in5,6,36,37.

For the following statement we refer to11.

Theorem A. LetX, d, μbe an SHT . Suppose that1< p− ≤p <∞andp∈ P1. Assume that

ifL∞, thenp≡const outside some ball. Letαbe a constant satisfying the condition0< α <1/p . One setsqx px/1−αpx. Then,Tαis bounded fromLp·XtoLq·X.

Theorem Bsee9. LetX, d, μbe a nonhomogeneous space withL <∞and letNbe a constant defined by N a11 2a0, where the constantsa0 and a1 are taken from the definition of the

quasimetricd. Suppose that1 < p₋ < p <∞, p, α ∈ PNand thatμis upper Ahlfors 1-regular. One definesqx px/1−αxpx, where0 < α− ≤α < 1/p . ThenIα· is bounded from Lp·XtoLq·X.

For the statements and their proofs of this section, we keep the notation of the previous sections and, in addition, introduce the new notation:

vα1x:vx

μBx0x α−1

, wα1x:w−1x; vα2x:vx;

wα2x:w−1x

μBx0x α−1

;

Fx:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y∈X: d

x0, y

L A2_a1 ≤d

x0, y

≤A2_La

1dx0, x

, ifL <∞,

y∈X: d

x0, y

A2_a1 ≤d

x0, y

≤A2_a

1dx0, x

, ifL∞,

4.1

where Aand a1 are constants defined inDefinition 2.11 and the triangle inequality ford,

Theorem 4.1. LetX, d, μbe an SHT without atoms. Suppose that1 < p− ≤ p < ∞andαis a

constant satisfying the condition0 < α < 1/p . Letp∈ P1. One setsqx px/1−αpx. Further, if L ∞, then one assumes that p ≡ pc ≡ const outside some ball Bx0, a. Then the

inequality

vTαf _Lq·_X≤c wf _Lp·_X 4.2

holds if the following three conditions are satisfied:

aT_v1

α ,wα1 is bounded fromL

p·_XtoLq·_X;

bT_v2

α ,wα2 is bounded fromL

p·_XtoLq·_X;

cthere is a positive constantbsuch that one of the following inequalities hold: (1)v Fx≤

bwxforμ−a.e.x∈X; (2) vx≤bw₋Fxforμ−a.e.x∈X.

Proof. For simplicity, suppose thatL < ∞. The proof for the caseL ∞is similar to that of the previous case. Recall that the setsIi,k, i1,2,3 andEkare defined inSection 2. Letf ≥0 and letg_Lq·_X≤1. We have

X

Tαf

xgxvxdμx

0

k−∞

Ek

Tαf

xgxvxdμx

≤ 0 k−∞

Ek

Tαf1,k

xgxvxdμx 0

k−∞

Ek

Tαf2,k

xgxvxdμx

0

k−∞

Ek

Tαf3,k

xgxvxdμx :S1 S2 S3,

4.3

wheref1,kf·χI1,k, f2,k f·χI2,k, f3,kf·χI3,k.

Observe that ifx ∈ Ekandy ∈ I1,k, thendx0, y ≤ dx0, x/Aa1. Consequently, the

triangle inequality fordyieldsdx0, x ≤ Aa1a0dx, y, whereA A/A−1. Hence, by

usingRemark 2.4, we find thatμBx0x≤cμBxy. Applying conditionanow, we have that

S1≤c

μBx0x

α−1 vx

Bx0x

fydμy

Lqx_X

_g

Lq·_X≤c f _Lp·_X. 4.4

Further, observe that ifx∈Ekandy ∈I3,k, thenμBx0y≤ cμBxy. By conditionb,

Now we estimateS2. Suppose thatv Fx≤bwx. Theorem A andLemma 2.14yield

S2≤

k

Tαf2,k

·χEk·v· _Lq·_X gχEk· _Lq·_X

≤

k

v Ek Tαf2,k

· _Lq·_X g·χEk· Lq·_X

≤c

k

v Ek f2,k _Lp·_X g·χEk· Lq·_X

≤c

k

f2,k·w·χI2,k· Lp·_X g·χEk· Lq·_X

≤c f·w· _Lp·_X g· _Lq·_X≤c f·w· _Lp·_X.

4.5

The estimate ofS2for the case whenvx≤bw−Fxis similar to that of the previous one. Details are omitted.

Theorems4.1,3.2, and3.3imply the following statement.

Theorem 4.2. LetX, d, μbe an SHT. Suppose that1< p₋≤p <∞andαis a constant satisfying the condition0 < α <1/p . Letp∈ P1. One setsqx px/1−αpx. IfL∞, then, one

supposes thatp≡pc ≡const outside some ballBx0, a. Then inequality4.2holds if the following

three conditions are satisfied:

i

P1 : sup 0<t≤L

t<dx0,x≤L

vx

μBx0x 1−α

qx

×

dx0,y≤t

w−p0x_ydμy

qx/p0x

dμx<∞;

4.6

ii

P2:sup 0<t≤L

dx0,x≤t

vxqx

×

t<dx0,y≤L

wyμBx0y

1−α−p1x dμy

qx/p1x

dμx<∞,

4.7

iiiconditioncofTheorem 4.1holds.

Remark 4.3. Ifp pc ≡const onX, then the conditionsPi < ∞,i 1,2, are necessary for

4.2. Necessity of the conditionP1 <∞follows by taking the test functionf w−pc

χBx0,t

in 4.2 and observing thatμBxy ≤ cμBx0x for thosex andy which satisfy the conditions dx0, x≥tanddx0, y≤tsee also31, Theorem 6.6.1, page 418for the similar arguments

fx w−pc_xχ

X\Bx0,txμBx0x

α−1pc−1 _{and taking into account the estimateμB} xy ≤

μBx0yfordx0, x≤tanddx0, y≥t.

The next statement follows in the same manner as the previous one. In this case, Theorem B is used instead of Theorem A. The proof is omitted.

Theorem 4.4. LetX, d, μbe a nonhomogeneous space withL <∞. LetNbe a constant defined by

Na11 2a0. Suppose that1< p−≤p <∞, p, α∈ PNand thatμis upper Ahlfors 1-regular.

We defineqx px/1−αxpx, where0< α−≤α <1/p . Then the inequality

v·Iα·f· Lq·_X≤c w·f· _Lp·_X 4.8

holds if

i

sup

0≤t≤L

t<dx0,x≤L

vx

dx0, x1−αx

qx

Bx0,t

w−p0x_ydμy

qx/p0x

dμx<∞;

4.9

ii

sup

0≤t≤L

Bx0,t

vxqx

t<dx0,y≤L

wydx0, y

1−αy−p1x dμy

qx/p1x

dμx<∞,

4.10

andiiiconditioncofTheorem 4.1is satisfied.

Remark 4.5. It is easy to check that ifp andαare constants, then conditionsiand ii in

Theorem 4.4 are also necessary for4.8. This follows easily by choosing appropriate test functions in4.8 see alsoRemark 4.3.

Theorem 4.6. LetX, d, μbe an SHT without atoms. Let1< p−≤p <∞and letαbe a constant

with the condition0< α <1/p . One setsqx px/1−αpx. Assume thatphas a minimum atx0and thatp∈LHX. Suppose also that ifL∞, thenpis constant outside some ballBx0, a.

Letvandwbe positive increasing functions on0,2L. Then the inequality

_vdx0,·

Tαf

· _Lq·_X≤c wdx0,·f· _Lp·_X 4.11

holds if

I1: sup 0<t≤L

I1t: sup 0<t≤L

t<dx0,x≤L

vdx0, x

μBx0x 1−α

qx

×

dx0,y≤t

w−p0x_dx 0, y

dμy

qx/p0x

dμx<∞,

forL∞;

J1: sup 0<t≤L

t<dx0,x≤L

vdx0, x

μBx0x 1−α

qx

×

dx0,y≤t

w−px0_dx 0, y

dμy

qx/px0

dμx<∞,

4.13

forL <∞.

Proof. We prove the theorem forL∞. The proof for the case whenL <∞is similar. Observe that by Lemma 2.10 the condition p ∈ LHX implies p ∈ P1. We will show that the conditionI1 < ∞ implies the inequalityvA2a1t/wt ≤ Cfor all t > 0, whereA anda1

are constants defined inDefinition 2.11and the triangle inequality ford, respectively. Indeed, let us assume thatt≤b1, whereb1is a small positive constant. Then, taking into account the

monotonicity ofvandwand the facts thatp0x p0x for smalldx0, xandμ∈RDCX,

we have

I1t≥

A2_a

1t≤dx0,x<A3a1t

vA2_a 1t

wt

qx

μBx0, t

α−1/p0xqx dμx

≥

vA2_a 1t

wt

q−

A2_{a1t≤dx0,x<A}3_a1t

μBx0, t

α−1/p0xqx

dμx≥c

vA2_a 1t

wt

q− .

4.14

Hence,c : limt→0vA2a1t/wt<∞. Further, ift > b2, whereb2 is a large number, then

sincepandqare constants, fordx0, x> t, we have that

I1t≥

A2_a

1t≤dx0,x<A3a1t

vdx0, xqc

μBx0, t

α−1qc dμx

×

Bx0,t

w−pcxdμx

qc/pc dμx

≥C

vA2_a 1t

wt

qc

A2_{a1t≤dx0,x<A}3_a1t

μBx0, t

α−1/pcqc_{dμx ≥c}

vA2_a 1t

wt

qc .

4.15

Now we show that the conditionI1<∞implies

sup t>0

I2t:sup

t>0

dx0,x≤t

vdx0, xqx

×

dx0,y>t

w−p1x_dx 0, y

μBx0y

α−1p1x_dμy

qx/p1x

dμx<∞.

4.16

Due to monotonicity of functionsvandw, the conditionp∈LHX,Proposition 2.6, Lemmas2.9, and2.10and the assumption thatphas a minimum atx0, we find that for all t >0,

I2t≤

dx0,x≤t

vt wt

qx

μBx0, t

α−1/px0qx_dμx

≤c

dx0,x≤t

vt wt

qx

μBx0, t

α−1/px0qx0_dμx

≤c

⎛

⎝

dx0,x≤t

vA2_a 1t

wt

qx dμx

⎞

⎠_{μBx0, t}−1_≤ C.

4.17

Now,Theorem 4.2completes the proof.

Theorem 4.7. LetX, d, μbe an SHT withL <∞. Suppose thatp,qandαare measurable functions onXsatisfying the conditions:1 < p− ≤px≤qx≤q <∞and1/p− < α− ≤α <1. Assume

thatα∈ LHXand there is a pointx0 ∈ X such thatp, q ∈LHX, x0. Suppose also thatwis a

positive increasing function on0,2L. Then the inequality

Tα·fv Lq·_X≤c wdx0,·f· _Lp·_X 4.18

holds if the following two conditions are satisfied:

I1: sup 0<t≤L

t≤dx0,x≤L

vx

μBx0x 1−αx

qx

×

dx0,x≤t

w−p0x_dx 0, y

dμy

qx/p0x

dμx<∞;

I2: sup 0<t≤L

dx0,x≤t

vxqx

×

t≤dx0,x≤L

wdx0, y

×μBx0y

1−αx−p1x dμy

qx/p1x

dμx<∞.

Proof. For simplicity, assume thatL 1. First observe that by Lemma 2.10we havep, q ∈

P1, x0andα ∈ P1. Suppose thatf ≥ 0 andSpwdx0,·f· ≤ 1. We will show that SqvTα·f≤C.

We have

Sq

vTα·f

≤Cq

⎡

⎣

X

vx

dx0,y≤dx0,x/2a1

fyμBxy

αx−1 dμy

qx dμx

X

vx

dx0,x/2a1≤dx0,y≤2a1dx0,x

fyμBxy

αx−1 dμy

qx dμx

X

vx

dx0,y≥2a1dx0,x

fyμBxy

αx−1 dμy

qx dμx

⎤

⎦:CqI1 I2 I3.

4.20

First, observe that by virtue of the doubling condition forμ,Remark 2.4, and simple calculation we find thatμBx0x≤cμBxy. Taking into account this estimate andTheorem 3.2

we have that

I1≤c

X

vx

μBx0x 1−αx

dx0,y<dx0,x

fydμy

qx

dμx≤C. 4.21

Further, it is easy to see that ifdx0, y ≥2a1dx0, x, then the triangle inequality for dand the doubling condition forμyield thatμBx0y ≤ cμBxy. Hence, due toProposition 2.7,

we see thatμBx0y

αx−1_≥cμB

xyαy−1for suchxandy. Therefore,Theorem 3.3implies that

I3≤C.

It remains to estimateI2. Let us denote:

E1_x:B

x0x\B

x0,

dx0, x

2a1

; E2_x:Bx

0,2a1dx0, x\Bx0x. 4.22

Then we have that

I2≤C

⎡

⎣

X

#

vx

E1_xf

yμBxy

αx−1 dμy

$qx dμx

X

#

vx

E2_xf

yμBxy

αx−1_dμy

$qx dμx

⎤

⎦:cI21 I22.