Fourier inequalities in variable Lebesgue spaces

Abstract

We study the boundedness of the Fourier transform on variable Lebesgue spaces. We obtain necessary conditions and, independently, sufficient conditions on the exponents $p(\cdot )$ and $q(\cdot )$ under which the inequality $$\Vert \hat{f}{\Vert }_{q(\cdot )}\le C\Vert f{\Vert }_{p(\cdot )}$$ holds for some constant $C>0$ and all $f\in L^{p(\cdot )}(\mathbb{R})$. As a byproduct, we improve some recent results of Saucedo and Tikhonov. Moreover, when $p(x)\to 1$ and $q(x)\to +\infty$ as $|x|\to +\infty$, we characterize the exponents for which the Fourier transform is bounded.