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http://dx.doi.org/10.4134/JKMS.j210747

FOURIER TRANSFORM OF ANISOTROPIC MIXED-NORM HARDY SPACES WITH APPLICATIONS TO HARDY-LITTLEWOOD INEQUALITIES  

Liu, Jun (School of Mathematics Chain University of Mining and Technology)
Lu, Yaqian (School of Mathematics Chain University of Mining and Technology)
Zhang, Mingdong (School of Mathematical Sciences Beijing Normal University)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.5, 2022 , pp. 927-944 More about this Journal
Abstract
Let $\vec{p}{\in}(0,\;1]^n$ be an n-dimensional vector and A a dilation. Let $H^{\vec{p}}_A(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H^{\vec{p}}_A(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f{\in}H^{\vec{p}}_A(\mathbb{R}^n)$ coincides with a continuous function F on ℝn in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H^{\vec{p}}_A(\mathbb{R}^n)$.
Keywords
Dilation; mixed-norm Hardy space; Fourier transform; Hardy-Littlewood inequality;
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