H\"ormander Type Functional Calculus and Square Function Estimates

TL;DR

This paper establishes H"ormander spectral multiplier theorems for elliptic operators on L^p spaces, linking them to square function estimates and extending results to operator space settings, with applications to semigroups with Gaussian bounds.

Contribution

It characterizes H"ormander multiplier theorems via square function estimates and extends the theory to matricial and operator space contexts for elliptic operators.

Findings

H"ormander multiplier theorems hold for a broad class of elliptic operators.

Square function estimates characterize the validity of these theorems.

Results apply to semigroups with generalized Gaussian estimates.

Abstract

We investigate H\"ormander 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 $\R^d.$ 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 H\"ormander 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 H\"ormander spectral multiplier theorems for semigroups satisfying generalized Gaussian estimates.