Semigroup Forum

, Volume 94, Issue 2, pp 260–296 | Cite as

Spectral multiplier theorems and averaged R-boundedness

Research Article

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 \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) 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 \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.

Keywords

Functional calculus Hörmander type spectral multiplier theorems R-boundedness Wave operators Imaginary powers 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.CNRS, LMBPUniversité Clermont AuvergneClermont-FerrandFrance
  2. 2.Institut für Analysis, Fakultät für MathematikKarlsruher Institut für TechnologieKarlsruheGermany

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