The Fourier Transform of Anisotropic Hardy Spaces with Variable Exponents and Their Applications

2021/12/21 by Wenhua Wang, Wang, Wenhua, Aiting Wang +1
Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2112.10956

Abstract

Let A be an expansive dilation on ℝⁿ, and p(⋅):ℝⁿ→(0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. Let H^p(⋅)_A(\mathbb Rⁿ) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors obtain that the Fourier transform of f∈ H^p(⋅)_A(\mathbb Rⁿ) coincides with a continuous function F on ℝⁿ in the sense of tempered distributions. As applications, the authors further conclude a higher order convergence of the continuous function F at the origin and then give a variant of the Hardy-Littlewood inequality in the setting of anisotropic Hardy spaces with variable exponents.

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