Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers

TL;DR

This paper proves the boundedness of Riesz transforms associated with the distinguished Laplacian on Damek-Ricci spaces for all p in (1,∞), using a novel operator-valued spectral multiplier theorem.

Contribution

It introduces a new operator-valued spectral multiplier theorem for joint functional calculus, enabling the proof of Riesz transform boundedness on Damek-Ricci spaces.

Findings

Proved L^p-boundedness of Riesz transforms for all p in (1,∞).

Developed an operator-valued spectral multiplier theorem of independent interest.

Extended the understanding of harmonic analysis on Damek-Ricci spaces.

Abstract

Let $\Delta = \nabla^* \nabla$ be the distinguished Laplacian on a Damek-Ricci space. We prove the $L^{p}$-boundedness of the vector of first-order Riesz transforms $\nabla \Delta^{-1/2}$ in the full range $p\in(1,\infty)$. The most demanding part of the proof is establishing the boundedness for $p \in (2,\infty)$; this is obtained as a consequence of an operator-valued spectral multiplier theorem for the joint functional calculus of a commuting system of self-adjoint operators, which we prove here and may be of independent interest.