Proceedings of the Japan Academy, Series A, Mathematical Sciences

Weighted inequalities for spherical maximal operator

Ramesh Manna

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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.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 9 (2015), 135-140.

Dates
First available in Project Euclid: 29 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1446124157

Digital Object Identifier
doi:10.3792/pjaa.91.135

Mathematical Reviews number (MathSciNet)
MR3418202

Zentralblatt MATH identifier
1334.42043

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords
Spherical maximal operator oscillatory integrals $A_{p}$ weights

Citation

Manna, Ramesh. Weighted inequalities for spherical maximal operator. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 9, 135--140. doi:10.3792/pjaa.91.135. https://projecteuclid.org/euclid.pja/1446124157


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