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 $$\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{q(\cdot )}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{p(\cdot )}$$ 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 $q(x)\to +\mathrm{\infty }$ as $\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }$, we characterize the exponents for which the Fourier transform is bounded.