Weighted bounds for the Carleson maximal operator in \(\mathbb{R}^ n\)
type: Item
instance of: scholarly article
title: Weighted bounds for the Carleson maximal operator in \(\mathbb{R}^ n\)
zbMATH Open document ID: 0816.42009
DOI: 10.1007/BF02844820
author: E. Romera
publication date: 1995-07-11T00:00:00Z
review text: Consider the Fourier multiplier (spherical partial sum operator) \(S_ R\) given by \((S_ R f)\sphat (\xi)= \chi_{\{|\xi|< R\}}\widehat f(\xi)\), and define the maximal operator \(T^*\) (Carleson maximal operator) by \(T^* f(x)= \sup_{R> 0}| S_ R f(x)|\). The author shows the following: Let \(1< p< \infty\) and let \(w(r)\) be a positive function with \(w(r) r^{(n-1)(1- p/2)}\in A_ p(\mathbb{R}^ +)\). Then there exists a constant \(C\) such that \[ \int_{\mathbb{R}^ n} | T^*(fP)(x)|^ p w(| x|) dx\leq C\int_{\mathbb{R}^ n} | f_ 0(| x|) P(x)|^ p w(| x|) dx \] for every radial function \(f(x)= f_ 0(| x|)\) and every homogeneous harmonic polynomial \(P(x)\). Here \(A_ p(\mathbb{R}^ +)\) coincides with the set of all even functions on \(\mathbb{R}\) which belong to the Muckenhoupt \(A_ p\) class . Unweighted and radial function case was treated by the authoress and \textit{F. Soria} in Proc. Am. Math. Soc. 111, No. 4, 1015- 1022 (1991; Zbl 0723.42004).
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 722326
zbMATH Keywords: weighted inequalities
zbMATH Keywords: spherical partial sum operator
zbMATH Keywords: Carleson maximal operator
zbMATH Keywords: Fourier multiplier
zbMATH Keywords: radial function
zbMATH Keywords: Muckenhoupt \(A_ p\) class
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
cites work: Weighted Inequalities for the Disc Multiplier
cites work: The multiplier for the ball and radial functions
cites work: The multiplier problem for the ball
cites work: Q3700300
cites work: ON THE MEAN INVERSION OF FOURIER AND HANKEL TRANSFORMS
cites work: Multiplier transformations. II
cites work: Convergence and Divergence Almost Everywhere of Spherical Means for Radial Functions
cites work: Almost everywhere convergence of the spherical partial sums for radial functions
cites work: Endpoint Estimates for the Maximal Operator Associated to Spherical Partial Sums on Radial Functions
cites work: Q5640160
cites work: Q5562752
published in: Rendiconti del Circolo Matematico di Palermo
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description: scientific article; zbMATH DE number 722326