Hörmander Type Functional Calculus and Square Function Estimates

Details Hörmander Type Functional Calculus and Square Function Estimates Hörmander Type Functional Calculus and Square Function Estimates

2012-01-23

arxiv.org/abs/1201.4830v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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