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Fourier transform of anisotropic mixed-norm Hardy spaces. (English) Zbl 1468.42023

Summary: Let \(\overrightarrow {a} : = (a_1, \ldots, a_n) \in [{1,\infty)^n, \overrightarrow{p}: = (p_1, \ldots, p_n) \in ({0,1}]^n},H_{\overrightarrow a}^{\overrightarrow p}\left( {{\mathbb{R}^n}} \right)\) 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 \(H_{\overrightarrow a}^{\overrightarrow p}\left( {{\mathbb{R}^n}} \right)\). In this article, we show that the Fourier transform \(\widehat{f}\) coincides with a continuous function \(g\) on \(\mathbb{R}^n\) 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 \(H_{\overrightarrow a}^{\overrightarrow p}\left({{\mathbb{R}^n}} \right)\) 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 \(H^p(\mathbb{R}^n)\) with \(p \in (0, 1]\), and are even new for isotropic mixed-norm Hardy spaces on \(\mathbb{R}^n\).

MSC:

42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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