Maximal functions and multiplier theorem for Fourier orthogonal series

TL;DR

This paper develops a maximal function and proves a multiplier theorem for Fourier orthogonal series on homogeneous spaces with addition formulas, extending harmonic analysis tools to conic domains.

Contribution

It introduces a convolution-based maximal function and establishes a Marcinkiewicz multiplier theorem for Fourier orthogonal expansions on conic domains.

Findings

Maximal function bounded by Hardy-Littlewood maximal function on conic domains

Multiplier theorem holds for Fourier orthogonal series on these domains

Extension of harmonic analysis techniques to multivariable orthogonal polynomials

Abstract

Under the assumption that orthogonal polynomials of several variables admit an addition formula, we can define a convolution structure and use it to study the Fourier orthogonal expansions on a homogeneous space. We define a maximal function via the convolution structure induced by the addition formula and use it to establish a Marcinkiewicz multiplier theorem. For the homogeneous space defined by a family of weight functions on conic domains, we show that the maximal function is bounded by the Hardy-Littlewood maximal function so that the multiplier theorem holds on conic domains.