Real-variable theory of Musielak-Orlicz Hardy spaces
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 the study of the weak Musielak-Orlicz Hardy space \(W H^{\varphi}(\mathbb{R}^n)\) and their characterizations, and the study of the local Musielak-Orlicz Hardy space \(h^{\varphi}(\mathbb{R}^n)\) establishing, in this case, the identification of its dual space \(\mathrm{BMO}^{\varphi}(\mathbb{R}^n)\). In Chapter 9, it is recalled the Musielak-Orlicz Besov and Triebel-Lizorkin spaces and it is given their \(\varphi\)-transform characterization in the sense of Frazier and Jawerth. As an application of the consideration of these spaces, in Chapter 10 it is proved that the product in the distribution sense of two functions, which are respectively in \(\mathrm{BMO}(\mathbb{R}^n)\) and \(H^1(\mathbb{R}^n)\), may be written as the sum of two continuous bilinear operators, which can be defined in terms of paraproducts one from \(H^1(\mathbb{R}^n) \times\mathrm{BMO}(\mathbb{R}^n)\) into \(L^1(\mathbb{R}^n)\) and the other one from \(H^1(\mathbb{R}^n) \times\mathrm{BMO}(\mathbb{R}^n)\) into a particular example of Musielak-Orlicz Hardy space with is \(H^{\log}(\mathbb{R}^n)\). Also, as an application of the study of these spaces, the final Chapter 11 deals with the boundedness of linear commutators \([b,T]\) from a suitable subspace \({\mathcal H}^1_b(\mathbb{R}^n)\) of \(H^1(\mathbb{R}^n)\), being \(b\) a function in \(\mathrm{BMO}(\mathbb{R}^n)\) and \(T\) a Calderón-Zygmund operator. Moreover, it is identified the largest subspace \( H^1_b(\mathbb{R}^n)\) such that all commutators of Calderón-Zygmund operators are continuous from \( H^1_b(\mathbb{R}^n)\) into \(L^1(\mathbb{R}^n)\), and some other endpoint estimates of the commutator \([b,T]\) are obtained with additional conditions on \(T\) and \(b\).
scientific article; zbMATH DE number 6696145