The Fourier transform on Rearrangement-Invariant Spaces

Abstract

We study inequalities of the form \begin{equation*} ρ( \lvert \hat{f} \rvert) \leq C σ(f) < \infty, \end{equation*} with $f\in L_1({\mathbb{R}}^n)$, the Lebesgue-integrable functions on ${\mathbb{R}}^n$ and \begin{equation*} \hat{f}(ξ) := \int_{\mathbb{R}^n} f(x) \, e^{- 2 πi ξ\cdot x} dx, \ \ \ ξ\in \mathbb{R}^n. \end{equation*} The functionals $\rho$ and $\sigma$ are so-called rearrangement-invariant (r.i.) norms on $M_{+}({\mathbb{R}}^n)$, the nonnegative measurable functions on ${\mathbb{R}}^n$. Results first proved in the general context of r.i. spaces are then both specialized and expanded on in the special cases of Orlicz spaces and of Lorentz Gamma spaces.