Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A→:=(A1,A2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec A: = \left( {{A_1},{A_2}} \right)$$\end{document} be a pair of expansive dilations and φ: ℝn×ℝm×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space HA→φ(ℝn×ℝm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$\end{document} via the anisotropic Lusin-area function and establish its atomic characterization, the g→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec g$$\end{document} -function characterization, the g→λ*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec g_\lambda ^*$$\end{document}-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of HA→φ(ℝn×ℝm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$\end{document} is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet ((φ,q,s→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\varphi ,q,\vec s} \right)$$\end{document}), if T is a sublinear operator and maps all ((φ,q,s→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\varphi ,q,\vec s} \right)$$\end{document})-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from HA→φ(ℝn×ℝm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$\end{document} to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from HA→φ(ℝn×ℝm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$\end{document} to Lφ(Rn × Rm) and from HA→φ(ℝn×ℝm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$\end{document} to itself, whose kernels are adapted to the action of A→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec A$$\end{document}. The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝn × ℝm and are new even for classical product Orlicz-Hardy spaces.