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

Abstract

Let $\Delta ={\nabla }^{\ast }\nabla$ 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.

52 pages