×

Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space \(L^{p(\cdot)}(\log L)^{q(\cdot)}\). (English) Zbl 1153.31002

Let \(p\) and \(q\) be two variable exponents on \({\mathbb R}^n\) satisfying certain log-Hölder conditions. In this paper, the authors study Riesz potentials of functions in generalized Lebesgue spaces \(L^{p(\cdot)}(\log L)^{q(\cdot)}\). The first main result of this paper is a Sobolev inequality for Riesz potentials of functions in \(L^{p(\cdot)}(\log L)^{q(\cdot)}\). The second main result of this paper says that for any non-negative function \(f\) on \({\mathbb R}^n\) with \(\| f\| _{L^{p(\cdot)}(\log L)^{q(\cdot)}({\mathbb R}^n)}\leq 1\) such that its Riesz potential is not identically infinite, every point in \({\mathbb R}^n\), except a set of zero capacity in some sense, is a Lebesgue point for the Riesz potential of \(f\). The third main result is on the vanishing exponential integrability of the potentials of functions in \(L^{p(\cdot)}(\log L)^{q(\cdot)}\).

MSC:

31C99 Generalizations of potential theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, D. R.; Hedberg, L. I., Function Spaces and Potential Theory (1996), Springer
[2] Adams, D. R.; Hurri-Syrjänen, R., Vanishing exponential integrability for functions whose gradients belong to \(L^n(\log(e + L))^\alpha \), J. Funct. Anal., 197, 162-178 (2003) · Zbl 1029.46021
[3] Almeida, A.; Samko, S., Characterization of Riesz and Bessel potentials on variable Lebesgue spaces, J. Funct. Spaces Appl., 4, 113-144 (2006) · Zbl 1129.46022
[4] D. Cruz-Uribe, A. Fiorenza, \(L \log L\) results for the maximal operator in variable \(L^p\) spaces, Trans. Amer. Math. Soc., in press; D. Cruz-Uribe, A. Fiorenza, \(L \log L\) results for the maximal operator in variable \(L^p\) spaces, Trans. Amer. Math. Soc., in press · Zbl 1163.42006
[5] Diening, L., Maximal functions in generalized \(L^{p(\cdot)}\) spaces, Math. Inequal. Appl., 7, 2, 245-254 (2004) · Zbl 1071.42014
[6] Diening, L., Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces \(L^{p(\cdot)}\) and \(W^{k, p(\cdot)}\), Math. Nachr., 268, 31-43 (2004) · Zbl 1065.46024
[7] Edmunds, D. E.; Rákosník, J., Sobolev embeddings with variable exponent, Studia Math., 143, 3, 267-293 (2000) · Zbl 0974.46040
[8] Fiorenza, A., A mean continuity type result for certain Sobolev spaces with variable exponent, Commun. Contemp. Math., 4, 3, 587-605 (2002) · Zbl 1015.46019
[9] Fiorenza, A.; Rakotoson, J. M., Relative rearrangement and Lebesgue spaces \(L^{p(\cdot)}\) with variable exponent, J. Math. Pures Appl., 88, 506-521 (2007) · Zbl 1137.46016
[10] Futamura, T.; Mizuta, Y., Continuity properties of Riesz potentials for functions in \(L^{p(\cdot)}\) of variable exponent, Math. Inequal. Appl., 8, 4, 619-631 (2005) · Zbl 1087.31004
[11] Futamura, T.; Mizuta, Y.; Shimomura, T., Sobolev embedding for Riesz potential spaces of variable exponent, Math. Nachr., 279, 1463-1473 (2006) · Zbl 1109.31004
[12] Futamura, T.; Mizuta, Y.; Shimomura, T., Sobolev embedding for variable exponent Riesz potentials on metric spaces, Ann. Acad. Sci. Fenn. Math., 31, 495-522 (2006) · Zbl 1100.31002
[13] Harjulehto, P.; Hästö, P., Lebesgue points in variable exponent spaces, Ann. Acad. Sci. Fenn. Math., 29, 295-306 (2004) · Zbl 1079.46022
[14] Harjulehto, P.; Hästö, P., A capacity approach to the Poincare inequality and Sobolev imbeddings in variable exponent Sobolev spaces, Rev. Mat. Complut., 17, 1, 129-146 (2004) · Zbl 1072.46021
[15] Harjulehto, P.; Hästö, P.; Koskenoja, M., Properties of capacities in variable exponent Sobolev spaces, J. Anal. Appl., 5, 2, 71-92 (2007) · Zbl 1143.31003
[16] Harjulehto, P.; Hästö, P.; Koskenoja, M.; Varonen, S., Sobolev capacity on the space \(W^{1, p(\cdot)}(R^n)\), J. Funct. Spaces Appl., 1, 1, 17-33 (2003) · Zbl 1078.46021
[17] Harjulehto, P.; Hästö, P.; Pere, M., Variable exponent Sobolev spaces on metric measure spaces, Funct. Approx. Comment. Math., 36, 79-94 (2006) · Zbl 1140.46013
[18] Hästö, P., The maximal operator in Lebesgue spaces with variable exponent near 1, Math. Nachr., 280, 1-2, 74-82 (2007) · Zbl 1125.46021
[19] Hedberg, L. I., On certain convolution inequalities, Proc. Amer. Math. Soc., 36, 505-510 (1972) · Zbl 0283.26003
[20] Kokilashvili, V.; Samko, S., On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent, Z. Anal. Anwend., 22, 4, 899-910 (2003) · Zbl 1040.42013
[21] Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\), Czechoslovak Math. J., 41, 592-618 (1991) · Zbl 0784.46029
[22] Meyers, N. G., A theory of capacities for potentials in Lebesgue classes, Math. Scand., 8, 255-292 (1970) · Zbl 0242.31006
[23] Meyers, N. G., Taylor expansion of Bessel potentials, Indiana Univ. Math. J., 23, 1043-1049 (1973/74) · Zbl 0288.31004
[24] Mizuta, Y., Potential Theory in Euclidean Spaces (1996), Gakkōtosyo: Gakkōtosyo Tokyo
[25] Mizuta, Y.; Ohno, T.; Shimomura, T., Integrability of maximal functions for generalized Lebesgue spaces with variable exponent, Math. Nachr., 281, 3, 386-395 (2008) · Zbl 1143.46010
[26] Mizuta, Y.; Shimomura, T., Maximal functions, Riesz potentials and Sobolev’s inequality in generalized Lebesgue spaces, (Potential Theory in Matsue. Potential Theory in Matsue, Adv. Stud. Pure Math., vol. 44 (2006), Math. Soc. Japan: Math. Soc. Japan Tokyo), 255-281 · Zbl 1125.31001
[27] Y. Mizuta, T. Shimomura, Vanishing exponential integrability for Riesz potentials of functions in Orlicz classes, Illinois J. Math., in press; Y. Mizuta, T. Shimomura, Vanishing exponential integrability for Riesz potentials of functions in Orlicz classes, Illinois J. Math., in press · Zbl 1152.31003
[28] Orlicz, W., Über konjugierte Exponentenfolgen, Studia Math., 3, 200-211 (1931) · Zbl 0003.25203
[29] Růžička, M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748 (2000), Springer · Zbl 0968.76531
[30] Samko, S.; Vakulov, B., Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl., 310, 229-246 (2005) · Zbl 1079.47053
[31] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0207.13501
[32] Trudinger, N., On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17, 473-483 (1967) · Zbl 0163.36402
[33] Ziemer, W. P., Extremal length as a capacity, Michigan Math. J., 17, 117-128 (1970) · Zbl 0183.39104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.