Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 551-571

WAVELET CHARACTERIZATION AND MODULAR INEQUALITIES FOR WEIGHTED LEBESGUE SPACES WITH VARIABLE EXPONENT

Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano

Okayama University, Graduate School of Education
Okayama 700-8530, Japan; izuki 'at' okayama-u.ac.jp

Ibaraki University, Department of Mathematics
Mito, Ibaraki 310-8512, Japan; enakai 'at' mx.ibaraki.ac.jp

Tokyo Metropolitan University, Department of Mathematics and Information Sciences
Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan; ysawano 'at' tmu.ac.jp

Abstract. In this paper, we characterize weighted Lebesgue spaces with variable exponent in terms of wavelet. Also, we disprove some weighted modular inequalities when the exponent is not a constant one without using the A_∞-condition on weights. As a byproduct, we shall obtain the vector-valued maximal inequalities in the weighted setting.

2010 Mathematics Subject Classification: Primary 42B35, 42C40.

Key words: Muckenhoupt weight, variable exponent, wavelet, weakly positive kernel, modular inequality.

Reference to this article: M. Izuki, E. Nakai and Y. Sawano: Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent. Ann. Acad. Sci. Fenn. Math. 40 (2015), 551-571.

Full document as PDF file

doi:10.5186/aasfm.2015.4032