Higher order Riesz transforms in the ultraspherical setting as principal value integral operators

TL;DR

This paper establishes the representation of higher order Riesz transforms in the ultraspherical setting as principal value integrals and analyzes their convergence speed through $L^p$ bounds on related oscillation and variation operators.

Contribution

It provides a novel representation of higher order ultraspherical Riesz transforms as principal value integrals and quantifies their convergence rates via $L^p$-boundedness of oscillation and variation operators.

Findings

Riesz transforms expressed as principal value integrals

Proved $L^p$-boundedness of oscillation operators

Quantified convergence speed of the transforms

Abstract

In this paper we represent the $k$-th Riesz transform in the ultraspherical setting as a principal value integral operator for every $k\in \mathbb{N}$. We also measure the speed of convergence of the limit by proving $L^p$-boundedness properties for the oscillation and variation operators associated with the corresponding truncated operators.