## Taiwanese Journal of Mathematics

### THE HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT AND THEIR APPLICATIONS

#### Abstract

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.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 4 (2012), 1363-1389.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406739

Digital Object Identifier
doi:10.11650/twjm/1500406739

Mathematical Reviews number (MathSciNet)
MR2951143

Zentralblatt MATH identifier
1260.42012

#### Citation

Wang, Hongbin; Liu, Zongguang. THE HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT AND THEIR APPLICATIONS. Taiwanese J. Math. 16 (2012), no. 4, 1363--1389. doi:10.11650/twjm/1500406739. https://projecteuclid.org/euclid.twjm/1500406739

#### 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.
• 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.
• L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math., 129 (2005), 657-700.
• 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.
• M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math., 36 (2010), 33-50.
• O. Kováčc ik and J. Rákosn\ipzz k, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
• S. Lu and D. Yang, The local versions of $H^p(\mathbb{R}^{n})$ spaces at the origin, Studia Math., 116 (1995), 103-131.
• S. Lu and D. Yang, The weighted Herz-type Hardy space and its applications, Sci. China Ser. A, 38 (1995), 662-673.
• A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(\mathbb{R}^{n})$, Math. Inequal. Appl., 7 (2004), 255-265.
• 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.