A Fourier inequality with \(A_ p\) and weak-\(L^ 1\) weight
type: Item
instance of: scholarly article
title: A Fourier inequality with \(A_ p\) and weak-\(L^ 1\) weight
zbMATH Open document ID: 0803.42013
published in: Collectanea Mathematica
publication date: 1995-01-03T00:00:00Z
full work available at URL: https://eudml.org/doc/42011
review text: 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).
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 633982
zbMATH Keywords: \(A_ p\)-weights
zbMATH Keywords: weak \(L^ 1\) weights
zbMATH Keywords: weighted Fourier norm inequality
zbMATH Keywords: radial weight function
author: H. P. Heinig
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 633982