In this paper, a certain Herz-type Hardy spaces with variable exponent are introduced, and characterizations of these spaces are established in terms of atomic and molecular decompositions. Using these decompositions, the authors obtain the boundedness of some operators on the Herz-type Hardy spaces with variable exponent.
References
D. Cruz-Uribe, SFO, A. Fiorenza and C. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223-238. 1064.42500 D. Cruz-Uribe, SFO, A. Fiorenza and C. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223-238. 1064.42500
D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. Pérez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 31 (2006), 239-264. D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. Pérez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 31 (2006), 239-264.
L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl., 7 (2004), 245-253. MR2057643 1071.42014 10.7153/mia-07-27 L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl., 7 (2004), 245-253. MR2057643 1071.42014 10.7153/mia-07-27
L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657-700. 1096.46013 10.1016/j.bulsci.2003.10.003 L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657-700. 1096.46013 10.1016/j.bulsci.2003.10.003
L. Diening, P. Harjulehto, P. Hästö and M. R\ipoožz ičc ka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., Vol. 2017, Springer-Verlag, Berlin, 2011. L. Diening, P. Harjulehto, P. Hästö and M. R\ipoožz ičc ka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., Vol. 2017, Springer-Verlag, Berlin, 2011.
P. Harjulehto, P. Hästö, Ú. V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. MR2639204 1188.35072 10.1016/j.na.2010.02.033 P. Harjulehto, P. Hästö, Ú. V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. MR2639204 1188.35072 10.1016/j.na.2010.02.033
M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math., 36 (2010), 33-50. MR2606575 1224.42025 10.1007/s10476-010-0102-8 M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math., 36 (2010), 33-50. MR2606575 1224.42025 10.1007/s10476-010-0102-8
S. Lu and D. Yang, The local versions of $H^p(\mathbb{R}^{n})$ spaces at the origin, Studia Math., 116 (1995), 103-131. 0935.42012 10.4064/sm-116-2-103-131 S. Lu and D. Yang, The local versions of $H^p(\mathbb{R}^{n})$ spaces at the origin, Studia Math., 116 (1995), 103-131. 0935.42012 10.4064/sm-116-2-103-131
S. Lu and D. Yang, The weighted Herz-type Hardy space and its applications, Sci. China Ser. A, 38 (1995), 662-673. 0832.42013 S. Lu and D. Yang, The weighted Herz-type Hardy space and its applications, Sci. China Ser. A, 38 (1995), 662-673. 0832.42013
L. Pick and M. R\ružička, An example of a space $L^{p(x)}$ on which the Hardy-Littlewood maximal operator is not bounded, Expo. Math., 19 (2001), 369-371. L. Pick and M. R\ružička, An example of a space $L^{p(x)}$ on which the Hardy-Littlewood maximal operator is not bounded, Expo. Math., 19 (2001), 369-371.
H. Wang, The decomposition for the Herz space with variable exponent and its applications, in: Proceedings of Academic Conference on Research Achievements of the Fundamental Research Funds for the Central Universities, China Coal Industry Publishing House, 2011. H. Wang, The decomposition for the Herz space with variable exponent and its applications, in: Proceedings of Academic Conference on Research Achievements of the Fundamental Research Funds for the Central Universities, China Coal Industry Publishing House, 2011.
