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Preprints, Working Papers, ... Year : 2014

Spectral multiplier theorems and R-boundedness

Abstract

Let $A$ be a $0$-sectorial operator with a bounded $H^\infty(\Sigma_\sigma)$-calculus for some $\sigma \in (0,\pi),$ e.g. a Laplace type operator on $L^p(\Omega),\: 1 < p < \infty,$ where $\Omega$ is a manifold or a graph. We show that $A$ has a $\Ha(\R_+)$ Hörmander functional calculus if and only if certain operator families derived from the resolvent $(\lambda - A)^{-1},$ the semigroup $e^{-zA},$ the wave operators $e^{itA}$ or the imaginary powers $A^{it}$ of $A$ are $R$-bounded in an $L^2$-averaged sense. If $X$ is an $L^p(\Omega)$ space with $1 \leq p < \infty,$ $R$-boundedness reduces to well-known estimates of square sums.
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Dates and versions

hal-01016676 , version 1 (30-06-2014)
hal-01016676 , version 2 (01-07-2014)
hal-01016676 , version 3 (23-10-2018)

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Christoph Kriegler, Lutz Weis. Spectral multiplier theorems and R-boundedness. 2014. ⟨hal-01016676v1⟩
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