Note on Fourier inequalities

TL;DR

This paper investigates Fourier inequalities in variable Lebesgue spaces, establishing conditions under which the Hausdorff--Young inequality holds and providing a full characterization of Pitt-type weighted inequalities.

Contribution

It proves that the Hausdorff--Young inequality in variable Lebesgue spaces holds only for constant exponents and characterizes Pitt-type inequalities with additional monotonicity assumptions.

Findings

Hausdorff--Young inequality holds only for constant p in variable Lebesgue spaces.

Full characterization of Pitt-type weighted Fourier inequalities with monotonicity.

Conditional results depend on the structure of the variable exponent p and weight functions.

Abstract

We prove that the Hausdorff--Young inequality $\|{\widehat{f}}\|_{q(\cdot)} \leq C \|{f}\|_{p(\cdot)}$ with $q(x)=p'(1/x)$ and $p(\cdot)$ even and non-decreasing holds in variable Lebesgue spaces if and only if $p$ is a constant. However, under the additional condition on monotonicity of $f$, we obtain a full characterization of Pitt-type weighted Fourier inequalities in the classical and variable Lebesgue setting.