Fourier transform of anisotropic mixed-norm Hardy spaces

Abstract

Let \overrightarrow{a}：=({{\displaystyle a}}_{\mathrm{1}},\mathrm{...},{{\displaystyle a}}_n)\in {[\mathrm{1},\infty )}^n,\overrightarrow{p}:=({{\displaystyle p}}_{\mathrm{1}},\mathrm{...}{{\displaystyle p}}_n)\in {(\mathrm{0},\mathrm{1}]}^n,{{\displaystyle H}}_{\overrightarrow{a}}^{\overrightarrow{p}}({\mathrm{ℝ}}^n) be the anisotropic mixed-norm Hardy space associated with \overrightarrow{a} defined via the radial maximal function, and let f belong to the Hardy space {{\displaystyle H}}_{\overrightarrow{a}}^{\overrightarrow{p}}({\mathrm{ℝ}}^n). In this article, we show that the Fourier transform \widehat{f} coincides with a continuous function g on {\mathrm{ℝ}}^n in the sense of tempered distributions and, moreover, this continuous function g; multiplied by a step function associated with \overrightarrow{a}; 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 {{\displaystyle H}}_{\overrightarrow{a}}^{\overrightarrow{p}}({\mathrm{ℝ}}^n) 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 H^p({\mathrm{ℝ}}^n) with p\in (\mathrm{0},\mathrm{1}], and are even new for isotropic mixed-norm Hardy spaces on {\mathrm{ℝ}}^n.

Key words： Anisotropic (mixed-norm) Hardy space; Fourier transform; Hardy-Littlewood inequality

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