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Sufficient Conditions for the Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

Commun. Math. Anal.
Volume 22, Number 2 (2019), 61 - 77

Sufficient Conditions for the Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

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Abstract

Let $f$ be an element of the subspace $(L^{p},l^{q})^{\alpha}(\mathbb{R}^d)$ ($1\leq p \leq \alpha \leq q \leq 2 $) of the Wiener amalgam space $(L^{p},l^{q})(\mathbb{R}^{d})$. We give sufficient conditions for Lebesgue integrability of the Fourier transform of $f.$ These conditions are in terms of the $(L^{p},l^{q})^{\alpha}(\mathbb{R}^{d})$ integral modulus of continuity of $f.$ As an application, we obtain that if $ 1\leq\alpha\leq q\le 2$ with $\frac{1}{\alpha}-\frac{1}{q}<\frac{1}{d}$ and $N= [\frac{d}{\alpha}] + 1,$ then the Fourier inversion theorem can be applied to the elements of the Sobolev space $ W^{N}((L^{1},l^{q})^{\alpha}(\mathbb{R}^d)).$