- Research Article
- Open Access
Potential Operators in Variable Exponent Lebesgue Spaces: Two-Weight Estimates
© Vakhtang Kokilashvili et al. 2010
- Received: 17 June 2010
- Accepted: 24 November 2010
- Published: 5 December 2010
Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces 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.
- Positive Constant
- Measurable Function
- Triangle Inequality
- Lebesgue Space
- Potential Operator
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 , 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., ). 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 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.,  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.
We need some definitions for the exponent which will be useful to derive the main results of the paper.
holds for all , . Further, if there are positive constants and such that (2.5) holds for all and all satisfying the condition .
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.
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., ).
It can be checked easily that if is an SHT, then .
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 .
Let be an and let . Then for all with , where is a small constant, and the constant does not depend on .
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.
if (resp., , then there are positive constants , , and such that for all and all (resp., for all with ), one has that .
Let , then there are positive constants , , and (in general, depending on ) such that for all ( ) and all one has .
Let , then there are positive constants , , and such that for all balls with radius ( ) and all , one has that .
Let be an SHT.
Part (i) was proved in  (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.
Let be an . Suppose that , then satisfies the condition (resp., ) if and only if resp., .
We follow .
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 .
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 .
A measure on is said to satisfy the reverse doubling condition if there exist constants and such that the inequality holds.
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]).
holds, where and the constant is independent of .
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 .
where the constants and are taken, respectively, from Definition 2.11 and the triangle inequality for the quasimetric , and is a diameter of .
for all and .
Moreover, the doubling condition yields , where . Hence, .
Moreover, using the doubling condition for we have that . This gives the estimates .
If we deal with a quasimetric measure space with , then we will assume that . Obviously, and in this case.
holds, then is bounded from to .
Finally, . Thus, is bounded if .
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 .
then is bounded from to .
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.
For the following statement we refer to .
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 ).
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 .
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.
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. .
Further, observe that if and , then . By condition (b), we find that .
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.
condition (c) of Theorem 4.1 holds.
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.
and (iii) condition (c) of Theorem 4.1 is satisfied.
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).
In the last inequality we used the fact that satisfies the reverse doubling condition.
Now, Theorem 4.2 completes the proof.
For simplicity, assume that . First observe that by Lemma 2.10 we have and . Suppose that and . We will show 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 .
Denote the first inner integral by and the second one by .
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.
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 4.4 completes the proof.
Let and , where and are constants satisfying the condition , . Then satisfies the conditions of Theorem 4.6.
The first and second authors were partially supported by the Georgian National Science Foundation Grant (project numbers: GNSF/ST09/23/3-100 and GNSF/ST07/3-169). A part of this work was fulfilled in Abdus Salam School of Mathematical sciences, GC University, Lahore. The second and third authors are grateful to the Higher Educational Commission of Pakistan for financial support. The authors express their gratitude to the referees for their very useful remarks and suggestions.
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