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

in weighted spaces which enable us to effectively construct examples of appropriate weights. The conditions are simultaneously necessary and sufficient for corresponding inequalities when the weights are of special type and the exponent of the space is constant. We assume that the exponent satisfies the local log-Hölder continuity condition, and if the diameter of is infinite, then we suppose that is 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 . As far as we know, unfortunately, an analog of the log-Hölder decay condition (at infinity) for is not known even in the unweighted case, which is well-known and natural for the Euclidean spaces (see [2–5]). Local log-Hölder continuity condition for the exponent , together with the log-Hölder decay condition, guarantees the boundedness of operators of harmonic analysis in spaces (see, 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 Calderón-Zygmund singular operators defined on .

Considerable interest of researchers is focused on the study of mapping properties of integral operators defined on (quasi)metric 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 in spaces was derived in the papers [7–14]. Weighted inequalities for classical operators in spaces, where is a power-type weight, were established in the papers [10–12, 15–19], while the same problems with general weights for Hardy, maximal, and fractional integral operators were studied in [10, 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 (which are also derived in this paper) and Hardy-Littlewood-Sobolev-type inequalities for and in spaces.

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

Finally we point out that constants (often different constants in the same series of inequalities) will generally be denoted by or . The symbol means that there are positive constants and independent of such that the inequality holds. Throughout the paper is denoted the function by the symbol .

Let be a topological space with a complete measure such that the space of compactly supported continuous functions is dense in and there exists a nonnegative real-valued function (quasimetric) on satisfying the conditions:

(i) if and only if ;

(ii)there exists a constant , such that for all ;

(iii)there exists a constant , such that for all .

We assume that the balls are measurable and for all and ; for every neighborhood of , there exists , such that . Throughout the paper we also suppose that and that

for all , positive and with , where

We call the triple a quasimetric measure space. If satisfies the doubling condition , where the positive constant does not depend on and , then is called a space of homogeneous type (SHT). For the definition, examples, and some properties of an SHT see, for example, monographs [28–30].

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

Notice that the condition implies that because we assumed that every ball in has a finite measure.

We say that the measure is upper Ahlfors -regular if there is a positive constant such that for for all and . Further, is lower Ahlfors -regular if there is a positive constant such that for all and . It is easy to check that if is a quasimetric measure space and , then is lower Ahlfors regular (see also, e.g., [8] for the case when is a metric).

For the boundedness of potential operators in weighted Lebesgue spaces with constant exponents on nonhomogeneous spaces we refer, for example, to the monograph [31, Chapter 6] and references cited therein.

Let be a nonnegative -measurable function on . Suppose that is a -measurable set in . We use the following notation:

Assume that . The variable exponent Lebesgue space (sometimes it is denoted by ) is the class of all -measurable functions on for which . The norm in is defined as follows:

It is known (see, e.g., [8, 15, 32, 33]) that is a Banach space. For other properties of spaces we refer, for example, to [32–34].

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

Definition 2.1.

Let be a quasimetric measure space and let be a constant. Suppose that satisfies the condition . We say that belongs to the class , where , if there are positive constants and (which might be depended on ) such that

holds for all , . Further, if there are positive constants and such that (2.5) holds for all and all satisfying the condition .

Definition 2.2.

Let be an SHT. Suppose that . We say that ( satisfies the log-Hölder-type condition at a point ) if there are positive constants and (which might be depended on ) such that

holds for all satisfying the condition . Further, ( satisfies the log-Hölder type condition on ) if there are positive constants and such that (2.6) holds for all with .

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

Definition 2.3.

Let be a quasimetric measure space, and let . We say that if there are positive constants and (which might be depended on ) such that

for all with . Further, if (2.7) holds for all with .

It is easy to see that if a measure is upper Ahlfors -regular and (resp., ), then (resp., . Further, if is lower Ahlfors -regular and (resp., ), then (resp., ).

Remark 2.4.

It can be checked easily that if is an SHT, then .

Remark 2.5.

Let be an SHT with . It is known (see, e.g., [8, 35]) that if , then . Further, if is upper Ahlfors -regular, then the condition implies that .

Proposition 2.6.

Let be positive and let and  (resp., , then the functions , and belong to resp., . Further if resp., then , and belong to resp., .

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

Proposition 2.7.

Let be an and let . Then for all with , where is a small constant, and the constant does not depend on .

Proof.

Due to the doubling condition for , Remark 1.1, the condition and the fact that we have the following estimates: , which proves the statement.

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

Proposition 2.8.

Let be a quasimetric measure space and let . Suppose that be a constant. Then the following statements hold:

(i) if (resp., , then there are positive constants , , and such that for all and all (resp., for all with ), one has that .

(ii) Let , then there are positive constants , , and (in general, depending on ) such that for all () and all one has .

(iii) Let , then there are positive constants , , and such that for all balls with radius () and all , one has that .

It is known that (see, e.g., [32, 33]) if is a measurable function on and is a measurable subset of , then the following inequalities hold:

Further, Hölder's inequality in the variable exponent Lebesgue spaces has the following form:

Lemma 2.9.

Let be an SHT.

(i)If is a measurable function on such that and if is a small positive number, then there exists a positive constant independent of and such that

(ii)Suppose that and are measurable functions on satisfying the conditions and . Then there exists a positive constant such that for all the inequality

holds.

Proof.

Part (i) was proved in [35] (see also [31, page 372], for constant ). The proof of Part (ii) is given in [31, (Lemma  6.5.2, page 348)] for constant and , but repeating those arguments we can see that it is also true for variable and . Details are omitted.

Lemma 2.10.

Let be an . Suppose that , then satisfies the condition (resp., ) if and only if resp., .

Proof.

We follow [1].

Necessity.

Let , and let with for some positive constant . Observe that , where . By the doubling condition for , we have that , where is a positive constant which is greater than 1. Taking now the logarithm in the last inequality, we have that . If , then by the same arguments we find that .

Sufficiency.

Let . First observe that If , then . Consequently, this inequality and the condition yield . Further, there exists such that and , where and are positive constants. Hence .

Let, now, and let where is a small number. We have that and for some positive constant . Consequently,

Definition 2.11.

A measure on is said to satisfy the reverse doubling condition if there exist constants and such that the inequality holds.

Remark 2.12.

It is known that if all annulus in are not empty (i.e., condition (2.1) holds), then implies that (see, e.g., [28, page 11, Lemma  20]).

Lemma 2.13.

Let be an . Suppose that there is a point such that . Let be the constant defined in Definition 2.11. Then there exist positive constants and (which might be depended on ) such that for all , the inequality

holds, where and the constant is independent of .

Proof.

Taking into account condition (2.1) and Remark 2.12, we have that . Let . By the doubling and reverse doubling conditions, we have that . Suppose that , where is a sufficiently small constant. Then by using Lemma 2.10 we find that .

In the sequel we will use the notation:

where the constants and are taken, respectively, from Definition 2.11 and the triangle inequality for the quasimetric , and is a diameter of .

Lemma 2.14.

Let be an and let . Suppose that there is a point such that . Assume that if , then and outside some ball . Then there exists a positive constant C such that

for all and .

Proof.

Suppose that . To prove the lemma, first observe that and . This holds because satisfies the reverse doubling condition and, consequently,

Moreover, the doubling condition yields , where . Hence, .

Further, since we can assume that , we find that

Moreover, using the doubling condition for we have that . This gives the estimates .

For simplicity, assume that . Suppose that is an integer such that . Let us split the sum as follows:

Since outside the ball , by using Hölder's inequality and the fact that , we have

Let us estimate . Suppose that and . Also, by Proposition 2.6, we have that . Therefore, by Lemma 2.13 and the fact that , we obtain that and , where . Further, observe that these estimates and Hölder's inequality yield the following chain of inequalities:

Now we claim that , where

and the positive constant does not depend on . Indeed, suppose that . Then taking into account Lemma 2.13 we have that

Consequently, since and , we find that

This implies that . Thus, the desired inequality is proved. Further, let us introduce the following function:

It is clear that because . Hence

for some positive constant . Then, by using this inequality, the definition of the function , the condition , and the obvious estimate , we find that

Consequently, . Hence, . Analogously taking into account the fact that and arguing as above, we find that . Thus, summarizing these estimates we conclude that

Lemma 2.14 for spaces defined with respect to the Lebesgue measure was derived in [24] (see also [22] for , , and ).

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

Let be a positive constant, and let be a measurable function defined on . Let us introduce the notation:

Remark 3.1.

If we deal with a quasimetric measure space with , then we will assume that . Obviously, and in this case.

Theorem 3.2.

Let be a quasimetric measure space. Assume that and are measurable functions on satisfying the condition . In the case when , suppose that const, const, outside some ball . If the condition

holds, then is bounded from to .

Proof.

Here we use the arguments of the proofs of Theorem  1.1.4 in [31, (see page 7)] and of Theorem  2.1 in [21]. First, we notice that for all . Let and let . First, assume that . We denote

Suppose that , then for some . Let us denote , and . Then is a nondecreasing sequence. It is easy to check that for , and . If , then if and only if . If , then we take . Since for every , we have that . It is obvious that . Further, we have that

Let us denote

Notice that . Consequently, by this estimate and Hölder's inequality with respect to the exponent we find that

where

Observe now that . Hence, this fact and the condition imply that

It follows now that

Since , it is obvious that

Finally, . Thus, is bounded if .

Let us now suppose that . We have

By using the already proved result for and the fact that , we find that because

Further, observe that

It is easy to see that (see also [31, Theorems  1.1.3 or  1.1.4]) the condition

guarantees the boundedness of the operator

from to . Thus, is bounded. It remains to prove that is bounded. We have

Observe, now, that the condition guarantees that the integral

is finite. Moreover, . Indeed, we have that

Further,

For , we have that . Since and condition (2.1) holds, there exists a point such that . Consequently, and , where . Consequently, the condition yields . Finally, we have that . Hence, is bounded from to .

The proof of the following statement is similar to that of Theorem 3.2; therefore, we omit it (see also the proofs of Theorem  1.1.3 in [31] and Theorems  2.6 and  2.7 in [21] for similar arguments).

Theorem 3.3.

Let be a quasimetric measure space. Assume that and are measurable functions on satisfying the condition . If , then, one assumes that const, const outside some ball . If

then is bounded from to .

Remark 3.4.

If const, then the condition in Theorem 3.2 (resp., in Theorem 3.3) is also necessary for the boundedness of (resp., ) from to . See [31, pages 4-5] for the details.

In this section, we discuss two-weight estimates for the potential operators and on quasimetric measure spaces, where . If , then we denote and by and , respectively.

The boundedness of Riesz potential operators in spaces, where is a domain in was established in [5, 6, 36, 37].

For the following statement we refer to [11].

Theorem A..

Let be an . Suppose that and . Assume that if , then outside some ball. Let be a constant satisfying the condition . One sets . Then, is bounded from to .

Theorem B (see [9]).

Let be a nonhomogeneous space with and let be a constant defined by , where the constants and are taken from the definition of the quasimetric . Suppose that and that is upper Ahlfors 1-regular. One defines , where . Then is bounded from to .

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

where and are constants defined in Definition 2.11 and the triangle inequality for , respectively. We begin this section with the following general-type statement.

Theorem 4.1.

Let be an SHT without atoms. Suppose that and is a constant satisfying the condition . Let . One sets . Further, if , then one assumes that const outside some ball . Then the inequality

holds if the following three conditions are satisfied:

(a) is bounded from to ;

(b) is bounded from to ;

(c)there is a positive constant such that one of the following inequalities hold: (1) for a.e. ; (2) for a.e. .

Proof.

For simplicity, suppose that . The proof for the case is similar to that of the previous case. Recall that the sets and are defined in Section 2. Let and let . We have

where .

Observe that if and , then . Consequently, the triangle inequality for yields , where . Hence, by using Remark 2.4, we find that . Applying condition (a) now, we have that

Further, observe that if and , then . By condition (b), we find that .

Now we estimate . Suppose that . Theorem A and Lemma 2.14 yield

The estimate of for the case when is similar to that of the previous one. Details are omitted.

Theorems 4.1, 3.2, and 3.3 imply the following statement.

Theorem 4.2.

Let be an SHT. Suppose that and is a constant satisfying the condition . Let . One sets . If , then, one supposes that const outside some ball . Then inequality (4.2) holds if the following three conditions are satisfied:

(i)

(ii)

(iii) condition (c) of Theorem 4.1 holds.

Remark 4.3.

If const on , then the conditions , , are necessary for (4.2). Necessity of the condition follows by taking the test function in (4.2) and observing that for those and which satisfy the conditions and (see also [31, Theorem  6.6.1, page 418] for the similar arguments) while necessity of the condition can be derived by choosing the test function and taking into account the estimate for and .

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.

Let be a nonhomogeneous space with . Let be a constant defined by . Suppose that and that is upper Ahlfors 1-regular. We define , where . Then the inequality

holds if

(i)

(ii)

and (iii) condition (c) of Theorem 4.1 is satisfied.

Remark 4.5.

It is easy to check that if and are constants, then conditions (i) and (ii) in Theorem 4.4 are also necessary for (4.8). This follows easily by choosing appropriate test functions in (4.8) (see also Remark 4.3).

Theorem 4.6.

Let be an SHT without atoms. Let and let be a constant with the condition . One sets . Assume that has a minimum at and that . Suppose also that if , then is constant outside some ball . Let and be positive increasing functions on . Then the inequality

holds if

for ;

for .

Proof.

We prove the theorem for . The proof for the case when is similar. Observe that by Lemma 2.10 the condition implies . We will show that the condition implies the inequality for all , where and are constants defined in Definition 2.11 and the triangle inequality for , respectively. Indeed, let us assume that , where is a small positive constant. Then, taking into account the monotonicity of and and the facts that (for small ) and , we have

Hence, . Further, if , where is a large number, then since and are constants, for , we have that

In the last inequality we used the fact that satisfies the reverse doubling condition.

Now we show that the condition implies

Due to monotonicity of functions and , the condition , Proposition 2.6, Lemmas 2.9 and 2.10, and the assumption that has a minimum at , we find that for all ,

Now, Theorem 4.2 completes the proof.

Theorem 4.7.

Let be an SHT with . Suppose that , and are measurable functions on satisfying the conditions: and . Assume that and there is a point such that . Suppose also that is a positive increasing function on . Then the inequality

holds if the following two conditions are satisfied:

Proof.

For simplicity, assume that . First observe that by Lemma 2.10 we have and . Suppose that and . We will show that .

We have

First, observe that by virtue of the doubling condition for , Remark 2.4, and simple calculation we find that . Taking into account this estimate and Theorem 3.2 we have that

Further, it is easy to see that if , then the triangle inequality for and the doubling condition for yield that . Hence, due to Proposition 2.7, we see that for such and . Therefore, Theorem 3.3 implies that .

It remains to estimate . Let us denote:

Then we have that

Using Hölder's inequality for the classical Lebesgue spaces we find that

Denote the first inner integral by and the second one by .

By using the fact that , where , we see that , while by applying Lemma 2.9, for , we have that

Summarizing these estimates for and we conclude that

By applying monotonicity of , the reverse doubling property for with the constants and (see Remark 2.12), and the condition we have that

Due to the facts that and is increasing, for , we find that

Analogously, the estimate for follows. In this case, we use the condition and the fact that when . The details are omitted. The theorem is proved.

Taking into account the proof of Theorem 4.6, we can easily derive the following statement, proof of which is omitted.

Theorem 4.8.

Let be an SHT with . Suppose that , and are measurable functions on satisfying the conditions and . Assume that . Suppose also that there is a point such that and has a minimum at . Let and be a positive increasing function on satisfying the condition (see Theorem 4.6). Then inequality (4.11) is fulfilled.

Theorem 4.9.

Let be an SHT with and let be upper Ahlfors 1-regular. Suppose that and that . Let have a minimum at . Assume that is constant satisfying the condition . We set . If and are positive increasing functions on satisfying the condition

then the inequality

holds.

Proof.

The proof is similar to that of Theorem 4.6, we only discuss some details. First, observe that due to Remark 2.5 we have that , where . It is easy to check that the condition implies that for all t, where the constant is defined in Definition 2.11 and is from the triangle inequality for . Further, Lemmas 2.9 and 2.10, the fact that has a minimum at , and the inequality

where the constant does not depend on and , yield that

Theorem 4.4 completes the proof.

Example 4.10.

Let and , where and are constants satisfying the condition , . Then satisfies the conditions of Theorem 4.6.