The Fourier Transform on Rearrangement-Invariant Spaces

Let \(\rho \) be a rearrangement-invariant (r.i.) norm on the set \(M({\mathbb {R}}^n)\) of Lebesgue-measurable functions on \({\mathbb {R}}^n\) such that the space \(L_{\rho }({\mathbb {R}}^n) = \left\{ f \in M({\mathbb {R}}^n): \rho (f) < \infty \right\} \) is an interpolation space between \(L_{2}({\mathbb {R}}^n)\) and \(L_{{\infty }}({\mathbb {R}}^n).\) The principal result of this paper asserts that given such a \(\rho ,\) the inequality

$$\begin{aligned} \rho ({\hat{f}}) \le C \sigma (f) \end{aligned}$$

holds for any r.i. norm \(\sigma \) on \( M({\mathbb {R}}^n)\) if and only if

$$\begin{aligned} {\bar{\rho }} \left( U f^{*} \right) \le C {\bar{\sigma }} (f^{*}). \end{aligned}$$

Here, \({\bar{\rho }}\) is the unique r.i. norm on \(M({\mathbb {R}}_+)\), \({\mathbb {R}}_+ = (0, \infty )\), satisfying \({\bar{\rho }}(f^{*})=\rho (f)\) and \(U f^{*} (t) = \int _{0}^{1/t} f^{*}\), in which \(f^{*}\) is the nonincreasing rearrangement of f on \(\mathbb {R_+}\). Further, in this case the smallest r.i. norm \(\sigma \) for which \(\rho ( {\hat{f}}) \le C \sigma (f)\) holds is given by

$$\begin{aligned} \sigma (f) = {\bar{\sigma }} (f^{*}) = {\bar{\rho }} \left( U f^{*}\right) , \end{aligned}$$

where, necessarily, \({\bar{\rho }} \left( \int _{0}^{1/t} \chi _{(0, a)} \right) = {\bar{\rho }} \left( \min \{1/t, \, a\} \right) < \infty \), for all \(a>0\). We further specialize and expand these results in the contexts of Orlicz and Lorentz Gamma spaces.