Higher order Riesz transforms in the inverse Gaussian setting and UMD Banach spaces

TL;DR

This paper investigates higher order Riesz transforms in the inverse Gaussian setting, establishing their boundedness on certain L^p spaces, and characterizes UMD Banach spaces through these transforms and related operators.

Contribution

It provides new boundedness results, principal value representations, and characterizations of UMD Banach spaces using higher order Riesz transforms in the inverse Gaussian context.

Findings

Boundedness of Riesz transforms on L^p spaces with inverse Gaussian measure

Representation of transforms as principal value singular integrals

Characterizations of UMD Banach spaces via Riesz transforms and imaginary powers

Abstract

In this paper we study higher order Riesz transforms associated with the inverse Gaussian measure given by $\pi ^{n/2}e^{|x|^2}dx$ on $\mathbb{R}^n$. We establish $L^p(\mathbb{R}^n,e^{|x|^2}dx)$-boundedness properties and obtain representations as principal values singular integrals for the higher order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.