Multivariate Spectral Multipliers

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.