Dimension free $L^p$ estimates for Riesz transforms via an $H^{\infty}$ joint functional calculus

TL;DR

This paper presents a dimension-free approach to establish $L^p$ bounds for Riesz transforms using an $H^{ olinebreak}^ ext{infty}$ joint functional calculus, applicable to various settings including orthogonal expansions and groups with polynomial growth.

Contribution

It introduces a novel scheme leveraging $H^{ olinebreak}^ ext{infty}$ calculus to derive dimension-independent $L^p$ bounds for Riesz transforms from one-dimensional cases.

Findings

Dimension-free $L^p$ bounds for Riesz transforms established.

Applicable to orthogonal expansions and discrete Riesz transforms on groups.

Provides a unified framework for multi-dimensional Riesz transform estimates.

Abstract

By using an $H^{\infty}$ joint functional calculus for strongly commuting operators, we derive a scheme to deduce the $L^p$ boundedness of certain $d$-dimensional Riesz transforms from the $L^p$ boundedness of appropriate one-dimensional Riesz transforms. Moreover, the $L^p$ bounds we obtain are independent of the dimension. The scheme is applied to Riesz transforms connected with orthogonal expansions and discrete Riesz transforms on products of groups with polynomial growth.