Fourier transform of anisotropic mixed-norm Hardy spaces

Let a→=(a1…an)∈[1∞)np→=(p1…pn)∈(01]nHa→p→(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overrightarrow a \,: = \,\left( {{a_1}, \ldots,{a_n}} \right) \in {\left[ {1,\infty } \right)^n},\,\overrightarrow {p\,}: = \left( {{p_1}, \ldots,{p_n}} \right) \in {\left( {0,1} \right]^n},H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)$$\end{document} be the anisotropic mixed-norm Hardy space associated with a→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overrightarrow a $$\end{document} defined via the radial maximal function, and let f belong to the Hardy space Ha→p→(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)$$\end{document}. In this article, we show that the Fourier transform f^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{f}$$\end{document} coincides with a continuous function g on ℝn in the sense of tempered distributions and, moreover, this continuous function g, multiplied by a step function associated with a→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overrightarrow a $$\end{document}, can be pointwisely controlled by a constant multiple of the Hardy space norm of f. These proofs are achieved via the known atomic characterization of Ha→p→(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)$$\end{document} and the establishment of two uniform estimates on anisotropic mixed-norm atoms. As applications, we also conclude a higher order convergence of the continuous function g at the origin. Finally, a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained. All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(ℝn) with p ∈ (0, 1], and are even new for isotropic mixed-norm Hardy spaces on ∈n.