A Fourier inequality with \(A_ p\) and weak-\(L^ 1\) weight (Q1332996)

The following weighted Fourier norm inequality is obtained: Let \(\omega\in A_ p\) be a radial weight function, which as radial function is non-decreasing in \((0,\infty)\). If \(\varphi\in L^ 1_{\text{weak}}\) and \(1< p\leq q\leq p'< \infty\), then the weighted Fourier inequality \[ \left\{\int_{\mathbb{R}^ n} | \widehat f(x)|^ q \omega\left({1\over | x|}\right)^{q/p} \varphi(x)^{1-q/p'} dx\right\}^{1/q}\leq C\left\{ \int_{\mathbb{R}^ n} | f(x)|^ p \omega(x)dx\right\}^{1/p} \] is satisfied. The cases \(\varphi(x)= | x|^{-n}\) and \(\omega(x)\equiv 1\) are known [cf. the author and \textit{G. J. Sinnamon}, Indiana Univ. Math. J. 38, No. 3, 603-628 (1989; Zbl 0668.42003), respectively \textit{L. Hörmander}, Acta Math. 104, 93-140 (1960; Zbl 0093.114).