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Journal of Operator Theory

Volume 71, Issue 1, Winter 2014  pp. 223-257.

Hoermander type functional calculus and square function estimates

Authors:  Christoph Kriegler
Author institution: Laboratoire de Mathematiques (CNRS UMR 6620), Universite Blaise-Pascal (Clermont-Ferrand 2), Campus des Cezeaux, 63177 Aubiere Cedex, France

Summary:  We investigate Hoermander 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 $A^{\mathrm it}$, for resolvents $R(\lambda,A),$ and for the analytic semigroup $\exp(-zA).$ We deduce Hoermander spectral multiplier theorems for semigroups satisfying generalized Gaussian estimates.

Keywords:  functional calculus, square functions, Hoermander spectral multipliers, operator spaces

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