On the unboundedness in $L^{q}$ of the Fourier transform of $L^{p}$ functions

Ahmed A. Abdelhakim

Published 2018-06-11Version 1

The Hausdorff-Young inequality confirms the continuity of the Fourier transform $\mathcal{F}$ from $L^{p}(\mathbb{R}^{d})$ to $L^{q}(\mathbb{R}^{d})$ exclusively when $1\leq p\leq 2$ and $q$ is its conjugate Lebesgue exponent. Let $\Omega$ be a Lebesgue measurable bounded subset of $\mathbb{R}^{d}$. We show that $\mathcal{F}$ is bounded from $L^{p}(\mathbb{R}^{d})$ to $L^{q}(\Omega)$ precisely when $\frac{1}{p}+\frac{1}{q}\geq 1$ and $1\leq p\leq 2$. We also prove that $\mathcal{F}$ is unbounded from $L^{p}(\mathbb{R}^{d})$ to $L^{q}(\mathbb{R}^{d}\setminus \Omega)$ outside the admissible range of the Hausdorff-Young inequality. This is the case even if $L^{p}(\mathbb{R}^{d})$ is replaced by the subspace of functions with compact support.