Some new inequalities for the Fourier transform for functions in generalized Lorentz spaces

The classical Hausdorff–Young and Hardy–Littlewood–Stein inequalities, relating functions on $\mathbb{R}$ and their Fourier transforms, are extended and complemented in various ways. In particular, a variant of the Hardy–Littlewood–Stein inequality covering the case $p\geqslant2$ is proved and two-sided estimates are derived.