Multivariate Spectral Multipliers

TL;DR

This thesis investigates conditions under which multivariate spectral multipliers for systems of commuting operators extend to bounded operators on L^p spaces, generalizing previous results and applying to various classical operators.

Contribution

It extends spectral multiplier theorems to multivariate systems of strongly commuting operators under heat semigroup contractivity assumptions.

Findings

Generalized spectral multiplier results to systems of operators.

Derived multivariate multiplier theorems for classical operators like Hermite and Laguerre.

Identified conditions for sharp boundedness of Riesz transforms.

Abstract

This thesis is devoted to the study of multivariate (joint) spectral multipliers for systems of strongly commuting non-negative self-adjoint operators, $L=(L_1,\ldots,L_d),$ on $L^2(X,\nu),$ where $(X,\nu)$ is a measure space. By strong commutativity we mean that the operators $L_r,$ $r=1,\ldots,d,$ admit a joint spectral resolution $E(\lambda).$ In that case, for a bounded function $m\colon [0,\infty)^d\to \mathbb{C},$ the multiplier operator $m(L)$ is defined on $L^2(X,\nu)$ by $$m(L)=\int_{[0,\infty)^d}m(\lambda)dE(\lambda).$$ By spectral theory, $m(L)$ is then bounded on $L^2(X,\nu).$ The purpose of the dissertation is to investigate under which assumptions on the multiplier function $m$ it is possible to extend $m(L)$ to a bounded operator on $L^p(X,\nu),$ $1<p<\infty.$ The crucial assumption we make is the $L^p(X,\nu),$ $1\leq p\leq \infty,$ contractivity of the heat semigroups corresponding to the operators $L_r,$ $r=1,\ldots,d.$ Under this assumption we generalize the results of [S. Meda, Proc. Amer. Math. Soc. 1990] to systems of strongly commuting operators. As an application we derive various multivariate multiplier theorems for particular systems of operators acting on separate variables. These include e.g. Ornstein-Uhlenbeck, Hermite, Laguerre, Bessel, Jacobi, and Dunkl operators. In some particular cases, we obtain presumably sharp results. Additionally, we demonstrate how a (bounded) holomorphic functional calculus for a pair of commuting operators, is useful in the study of dimension free boundedness of various Riesz transforms.