Martin Mazzitelli

Fourier inequalities in variable Lebesgue spaces

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 $$\|\hat{f}\|_{q(\cdot)}\leq C\|f\|_{p(\cdot)}$$ 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.