Hörmander Type Functional Calculus and Square Function Estimates

Abstract

We investigate Hörmander spectral multiplier theorems as they hold on $X=L^p(\Omega ),1<p<\infty ,$ for many self-adjoint elliptic differential operators $A$ including the standard Laplacian on ${\mathbb{R}}^d.$ A strengthened matricial extension is considered, which coincides with a completely bounded map between operator spaces in the case that $X$ is a Hilbert space. We show that the validity of the matricial Hörmander theorem can be characterized in terms of square function estimates for imaginary powers $A^{it}$, for resolvents $R(\lambda ,A),$ and for the analytic semigroup $\exp (-zA).$ We deduce Hörmander spectral multiplier theorems for semigroups satisfying generalized Gaussian estimates.