Spectral multiplier theorems and averaged 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 H{ö}rmander 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.

Error in the title corrected

References in corpus (1)

• Spectral multipliers for wave operators