Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2021.12.10
  • Accepted : 2022.07.11
  • Published : 2022.09.01
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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

Acknowledgement

This research was financially supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20200647), the National Natural Science Foundation of China (Grant No. 12001527) and the Project Funded by China Postdoctoral Science Foundation (Grant No. 2021M693422).

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