Real-variable theory of Musielak-Orlicz Hardy spaces
type: Item
instance of: scholarly article
title: Real-variable theory of Musielak-Orlicz Hardy spaces
zbMATH Open document ID: 1375.42038
DOI: 10.1007/978-3-319-54361-1
author: Dachun Yang
author: Luong Dang Ky
author: Yiyu Liang
published in: Lecture Notes in Mathematics
publication date: 2017-03-20T00:00:00Z
review text: As it is well known in harmonic analysis, real Hardy spaces \(H^p(\mathbb{R}^n)\) with \(p\in (0,1]\) represent a substitute of the Lebesgue spaces \(L^p(\mathbb{R}^n)\) when studying the boundedness of some operators. Similarly, Musielak-Orlicz Hardy spaces are also substitutes of Musielak-Orlicz spaces when dealing with structures defined via the growth functions that may vary in both the spatial variable and the growth variable. The main purpose of the book is to give a complete survey of the different characterizations of the Musielak-Orlicz Hardy spaces \(H^{\varphi}(\mathbb{R}^n)\), where \(\varphi: \mathbb{R}^n\times [0,+\infty)\longrightarrow [0,+\infty)\) is a function such that \(\varphi(x,\cdot)\) is an Orlicz function and \(\varphi(\cdot, t)\) is a Muckenhoupt-\(A_{\infty}\) weight. Starting with a non-tangential grand maximal characterization of \(H^{\varphi}(\mathbb{R}^n)\), an atomic characterization is established in Chapter 1. Also, the identification of the dual of these spaces corresponding to certain growth functions \(\varphi\), as \(\mathrm{BMO}^{\varphi}(\mathbb{R}^n)\), is obtained in the same chapter. Chapter 2 through 4 are devoted to establishing some new real variable characterizations of \(H^{\varphi}(\mathbb{R}^n)\) such as maximal characterization of type vertical and non-tangential, a characterization in terms of Littlewood-Paley functions, and in terms of molecular decompositions. Moreover, it is established a Riesz transform characterization of \(H^{\varphi}(\mathbb{R}^n)\) by means of some higher order Riesz transforms. Chapter 5 deals with duality results. In particular, it is shown that some particular cases of Musielak-Orlicz Campanato spaces \({\mathcal L}_{\varphi,q,s}(\mathbb{R}^n)\) are the dual space of Musielak-Orlicz Hardy spaces \(H^{\varphi}(\mathbb{R}^n)\) . In Chapter 6 they are established the intrinsic square function characterization of \(H^{\varphi}(\mathbb{R}^n)\) . Chapters 7 and 8 are respectively devoted to th…
reviewed by: Santiago Boza
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 6696145
zbMATH Keywords: Musielak-Orlicz spaces
zbMATH Keywords: Hardy spaces
zbMATH Keywords: BMO
zbMATH Keywords: Campanato spaces
zbMATH Keywords: Calderón-Zygmund operators
zbMATH Keywords: commutators
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-54361-1
OpenAlex ID: W2612029172
description: scientific article; zbMATH DE number 6696145