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Journal Articles Journal of Operator Theory Year : 2014

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 $\R^d.$ 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.
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Dates and versions

hal-00662259 , version 1 (23-01-2012)

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Christoph Kriegler. Hörmander Type Functional Calculus and Square Function Estimates. Journal of Operator Theory, 2014, 71 (1), pp.223-257. ⟨hal-00662259v1⟩
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