The Fourier Transform on Rearrangement-Invariant Spaces

Let \(\rho \) $\rho$ be a rearrangement-invariant (r.i.) norm on the set \(M({\mathbb {R}}^n)\) $M({\mathbb{R}}^n)$ of Lebesgue-measurable functions on \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ such that the space \(L_{\rho }({\mathbb {R}}^n) = \left\{ f \in M({\mathbb {R}}^n): \rho (f) < \infty \right\} \) $L_{\rho }({\mathbb{R}}^n)=\left\{f,\in ,M,({\mathbb{R}}^n),:,\rho ,(f),<,\infty \right\}$ is an interpolation space between \(L_{2}({\mathbb {R}}^n)\) $L_2({\mathbb{R}}^n)$ and \(L_{{\infty }}({\mathbb {R}}^n).\) $L_{\infty }({\mathbb{R}}^n).$ The principal result of this paper asserts that given such a \(\rho ,\) $\rho ,$ the inequality \(\begin{aligned} \rho ({\hat{f}}) \le C \sigma (f) \end{aligned}\) $$\begin{array}{c}\hfill \rho (\widehat{f})\le C\sigma (f)\end{array}$$ holds for any r.i. norm \(\sigma \) $\sigma$ on \( M({\mathbb {R}}^n)\) $M({\mathbb{R}}^n)$ if and only if \(\begin{aligned} {\bar{\rho }} \left( U f^{*} \right) \le C {\bar{\sigma }} (f^{*}). \end{aligned}\) $$\begin{array}{c}\hfill \overline{\rho }\left(U,f_{}^{\ast }\right)\le C\overline{\sigma }(f_{}^{\ast }).\end{array}$$ Here, \({\bar{\rho }}\) $\overline{\rho }$ is the unique r.i. norm on \(M({\mathbb {R}}_+)\) $M({\mathbb{R}}_{+})$ , \({\mathbb {R}}_+ = (0, \infty )\) ${\mathbb{R}}_{+}=(0,\infty )$ , satisfying \({\bar{\rho }}(f^{*})=\rho (f)\) $\overline{\rho }(f_{}^{\ast })=\rho (f)$ and \(U f^{*} (t) = \int _{0}^{1/t} f^{*}\) $Uf_{}^{\ast }(t)={\int }_0^{1/t}f_{}^{\ast }$ , in which \(f^{*}\) $f_{}^{\ast }$ is the nonincreasing rearrangement of f on \(\mathbb {R_+}\) ${\mathbb{R}}_{+}$ . Further, in this case the smallest r.i. norm \(\sigma \) $\sigma$ for which \(\rho ( {\hat{f}}) \le C \sigma (f)\) $\rho (\widehat{f})\le C\sigma (f)$ holds is given by \(\begin{aligned} \sigma (f) = {\bar{\sigma }} (f^{*}) = {\bar{\rho }} \left( U f^{*}\right) , \end{aligned}\) $$\begin{array}{c}\hfill \sigma (f)=\overline{\sigma }(f_{}^{\ast })=\overline{\rho }\left(U,f_{}^{\ast }\right),\end{array}$$ where, necessarily, \({\bar{\rho }} \left( \int _{0}^{1/t} \chi _{(0, a)} \right) = {\bar{\rho }} \left( \min \{1/t, \, a\} \right) < \infty \) $\overline{\rho }\left({\int }_0^{1/t},{\chi }_{(0,a)}\right)=\overline{\rho }\left(min,\{,1,/,t,,,,a,\}\right)<\infty$ , for all \(a>0\) $a>0$ . We further specialize and expand these results in the contexts of Orlicz and Lorentz Gamma spaces.