## Banach Journal of Mathematical Analysis

### Wavelet characterizations of Musielak–Orlicz Hardy spaces

#### Abstract

In this article, via establishing a new atomic characterization of the Musielak–Orlicz Hardy space $H^{\varphi}({\mathbb{R}}^{n})$ [which is essentially deduced from the known molecular characterization of $H^{\varphi}({\mathbb{R}}^{n})$] and some estimates on a new discrete Littlewood–Paley $g$-function and a Peetre-type maximal function, together with using the known intrinsic $g$-function characterization of $H^{\varphi}({\mathbb{R}}^{n})$, the authors obtain several equivalent characterizations of $H^{\varphi}({\mathbb{R}}^{n})$ in terms of wavelets, which extend the wavelet characterizations of both Orlicz–Hardy spaces and the weighted Hardy spaces, and are available to the typical and useful Musielak–Orlicz Hardy space $H^{\log}({\mathbb{R}}^{n})$. The novelty of this approach is that the new adapted atomic characterization of $H^{\varphi}({\mathbb{R}}^{n})$ compensates the inconvenience in applications of the supremum appearing in the original definition of atoms, which play crucial roles in the proof of the main theorem of this article and may have further potential applications.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 1017-1046.

Dates
Accepted: 26 March 2018
First available in Project Euclid: 4 September 2018

https://projecteuclid.org/euclid.bjma/1536048016

Digital Object Identifier
doi:10.1215/17358787-2018-0010

Mathematical Reviews number (MathSciNet)
MR3858759

Zentralblatt MATH identifier
06946301

#### Citation

Fu, Xing; Yang, Dachun. Wavelet characterizations of Musielak–Orlicz Hardy spaces. Banach J. Math. Anal. 12 (2018), no. 4, 1017--1046. doi:10.1215/17358787-2018-0010. https://projecteuclid.org/euclid.bjma/1536048016

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