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.
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:
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.
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 will also need another form of the log-Hölder continuity condition given by the following definition.
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.
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 .
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]).
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 .
3. Hardy-Type Transforms
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 .
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.
For the following statement we refer 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 .
holds if the following three conditions are satisfied:
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).
Now, Theorem 4.2 completes the proof.
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 .
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.
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.
- Diening L: Maximal function on generalized Lebesgue spaces . Mathematical Inequalities & Applications 2004, 7(2):245–253.MathSciNetView ArticleMATHGoogle Scholar
- Cruz-Uribe D, Fiorenza A, Neugebauer CJ: The maximal function on variable spaces. Annales Academiae Scientiarum Fennicae Mathematica 2003, 28(1):223–238.MathSciNetMATHGoogle Scholar
- Cruz-Uribe D, Fiorenza A, Neugebauer CJ: Erratum: The maximal function on variable spaces. Annales Academiae Scientiarum Fennicae Mathematica 2004, 29(1):247–249.MathSciNetMATHGoogle Scholar
- Nekvinda A: Hardy-Littlewood maximal operator on . Mathematical Inequalities & Applications 2004, 7(2):255–265.MathSciNetView ArticleMATHGoogle Scholar
- Capone C, Cruz-Uribe D, Fiorenza A: The fractional maximal operator and fractional integrals on variable spaces. Revista Mathemática. Iberoamericana 2007, 23(3):743–770.MathSciNetView ArticleMATHGoogle Scholar
- Cruz-Uribe D, Fiorenza A, Martell JM, Pérez C: The boundedness of classical operators on variable spaces. Annales Academiæ Scientiarium Fennicæ. Mathematica 2006, 31(1):239–264.MathSciNetMATHGoogle Scholar
- Harjulehto P, Hästö P, Latvala V: Sobolev embeddings in metric measure spaces with variable dimension. Mathematische Zeitschrift 2006, 254(3):591–609. 10.1007/s00209-006-0960-8MathSciNetView ArticleMATHGoogle Scholar
- Harjulehto P, Hästö P, Pere M: Variable exponent Lebesgue spaces on metric spaces: the Hardy-Littlewood maximal operator. Real Analysis Exchange 2004/05, 30(1):87–103.MathSciNetMATHGoogle Scholar
- Kokilashvili V, Meskhi A: Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure. Complex Variables and Elliptic Equations 2010, 55(8–10):923–936.MathSciNetView ArticleMATHGoogle Scholar
- Kokilashvili V, Samko S: The maximal operator in weighted variable spaces on metric measure spaces. Proceedings of A. Razmadze Mathematical Institute 2007, 144: 137–144.MathSciNetMATHGoogle Scholar
- Kokilashvili VM, Samko SG: Operators of harmonic analysis in weighted spaces with non-standard growth. Journal of Mathematical Analysis and Applications 2009, 352(1):15–34. 10.1016/j.jmaa.2008.06.056MathSciNetView ArticleMATHGoogle Scholar
- Kokilashvili V, Samko S: The maximal operator in weighted variable exponent spaces on metric spaces. Georgian Mathematical Journal 2008, 15(4):683–712.MathSciNetMATHGoogle Scholar
- Khabazi M: Maximal functions in spaces. Proceedings of A. Razmadze Mathematical Institute 2004, 135: 145–146.MathSciNetMATHGoogle Scholar
- Almeida A, Samko S: Fractional and hypersingular operators in variable exponent spaces on metric measure spaces. Mediterranean Journal of Mathematics 2009, 6(2):215–232. 10.1007/s00009-009-0006-7MathSciNetView ArticleMATHGoogle Scholar
- Kokilashvili V, Samko S: Maximal and fractional operators in weighted spaces. Revista Matemática Iberoamericana 2004, 20(2):493–515.MathSciNetView ArticleMATHGoogle Scholar
- Kokilashvili V, Samko S: On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Zeitschrift für Analysis und ihre Anwendungen 2003, 22(4):899–910.MathSciNetView ArticleMATHGoogle Scholar
- Samko S, Vakulov B: Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. Journal of Mathematical Analysis and Applications 2005, 310(1):229–246. 10.1016/j.jmaa.2005.02.002MathSciNetView ArticleMATHGoogle Scholar
- Samko NG, Samko SG, Vakulov BG: Weighted Sobolev theorem in Lebesgue spaces with variable exponent. Journal of Mathematical Analysis and Applications 2007, 335(1):560–583. 10.1016/j.jmaa.2007.01.091MathSciNetView ArticleMATHGoogle Scholar
- Diening L, Samko S: Hardy inequality in variable exponent Lebesgue spaces. Fractional Calculus & Applied Analysis 2007, 10(1):1–18.MathSciNetMATHGoogle Scholar
- Edmunds DE, Kokilashvili V, Meskhi A: A trace inequality for generalized potentials in Lebesgue spaces with variable exponent. Journal of Function Spaces and Applications 2004, 2(1):55–69.MathSciNetView ArticleMATHGoogle Scholar
- Edmunds DE, Kokilashvili V, Meskhi A: On the boundedness and compactness of weighted Hardy operators in spaces . Georgian Mathematical Journal 2005, 12(1):27–44.MathSciNetMATHGoogle Scholar
- Edmunds DE, Kokilashvili V, Meskhi A: Two-weight estimates in spaces with applications to Fourier series. Houston Journal of Mathematics 2009, 35(2):665–689.MathSciNetMATHGoogle Scholar
- Kokilashvili V, Meskhi A: Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent. Integral Transforms and Special Functions 2007, 18(9–10):609–628.MathSciNetView ArticleMATHGoogle Scholar
- Kopaliani TS: On some structural properties of Banach function spaces and boundedness of certain integral operators. Czechoslovak Mathematical Journal 2004, 54(129)(3):791–805.MathSciNetView ArticleMATHGoogle Scholar
- Diening L, Hästö P: Muckenhoupt weights in variable exponent spaces. preprint, http://www.helsinki.fi/~pharjule/varsob/publications.shtml
- Kokilashvili V, Krbec M: Weighted inequalities in Lorentz and Orlicz spaces. World Scientific, River Edge, NJ, USA; 1991:xii+233.View ArticleMATHGoogle Scholar
- Kokilashvili V: New aspects in the weight theory and applications. In Function Spaces, Differential Operators and Nonlinear Analysis (Paseky nad Jizerou, 1995). Edited by: Krbec Met al.. Prometheus, Prague, Czech Republic; 1996:51–70.Google Scholar
- Strömberg J-O, Torchinsky A: Weighted Hardy Spaces, Lecture Notes in Mathematics. Volume 1381. Springer, Berlin, Germany; 1989:vi+193.MATHGoogle Scholar
- Coifman RR, Weiss G: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Mathematics. Volume 242. Springer, Berlin, Germany; 1971:v+160.Google Scholar
- Folland GB, Stein EM: Hardy Spaces on Homogeneous Groups, Mathematical Notes. Volume 28. Princeton University Press, Princeton, NJ, USA; 1982:xii+285.Google Scholar
- Edmunds DE, Kokilashvili V, Meskhi A: Bounded and Compact Integral Operators, Mathematics and Its Applications. Volume 543. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002:xvi+643.View ArticleMATHGoogle Scholar
- Kováčik O, Rákosník J: On spaces and . Czechoslovak Mathematical Journal 1991, 41(116)(4):592–618.MATHGoogle Scholar
- Samko SG: Convolution type operators in . Integral Transforms and Special Functions 1998, 7(1–2):123–144. 10.1080/10652469808819191MathSciNetView ArticleMATHGoogle Scholar
- Sharapudinov II: The topology of the space . Matematicheskie Zametki 1979, 26(4):613–632, 655.MathSciNetMATHGoogle Scholar
- Kokilashvili V, Meskhi A: Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Armenian Journal of Mathematics 2008, 1(1):18–28.MathSciNetMATHGoogle Scholar
- Diening L: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces and . Mathematische Nachrichten 2004, 268: 31–43. 10.1002/mana.200310157MathSciNetView ArticleMATHGoogle Scholar
- Samko SG: Convolution and potential type operators in . Integral Transforms and Special Functions 1998, 7(3–4):261–284. 10.1080/10652469808819204MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.