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November 2015 Weighted inequalities for spherical maximal operator
Ramesh Manna
Proc. Japan Acad. Ser. A Math. Sci. 91(9): 135-140 (November 2015). DOI: 10.3792/pjaa.91.135

Abstract

Given a set $E=(0, \infty)$, the spherical maximal operator $\mathcal{M}$ associated to the parameter set $E$ is defined as the supremum of the spherical means of a function when the radii of the spheres are in $E$. The aim of this paper is to study the following inequality \begin{equation} ∫_{\mathbf{R}^{n}} (\mathcal{M}f(x))^{p} φ(x) dx ≤ B_{p} ∫_{\mathbf{R}^{n}} |f(x)|^{p} φ(x) dx, \label{Lb1} \end{equation} holds for $p > \frac{2n}{n-1}$ with the continuous spherical maximal operator $\mathcal{M}$ and where the nonnegative function $\phi$ is in some weights obtained from the $A_{p}$ classes. As an application, we will get the boundedness of vector-valued extension of the spherical means.

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Ramesh Manna. "Weighted inequalities for spherical maximal operator." Proc. Japan Acad. Ser. A Math. Sci. 91 (9) 135 - 140, November 2015. https://doi.org/10.3792/pjaa.91.135

Information

Published: November 2015
First available in Project Euclid: 29 October 2015

zbMATH: 1334.42043
MathSciNet: MR3418202
Digital Object Identifier: 10.3792/pjaa.91.135

Subjects:
Primary: 42B25
Secondary: 42B20

Keywords: $A_{p}$ weights , Oscillatory integrals , Spherical maximal operator

Rights: Copyright © 2015 The Japan Academy

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Vol.91 • No. 9 • November 2015
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