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

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 $\frac{1}{\alpha}-\frac{1}{q}\lt\frac{1}{d}$ and $N= [\frac{d}{\alpha}] + 1$, then the Fourier inversion theorem can be applied to the elements of the Sobolev space $W^{N}((L^{1},l^{q})^{\alpha}(\mathbb{R}^d))$.