$H^\infty$-Calculus for cylindrical boundary value problems

Abstract

In this note an ${\mathcal{R}}$-bounded ${\mathcal{H}}^\infty$-calculus for linear operators associated to cylindrical boundary value problems is proved. The obtained results are based on an abstract result on operator-valued functional calculus by N. Kalton and L. Weis; cf. [28]. Cylindrical in this context means that both domain and differential operator possess a certain cylindrical structure. In comparison to standard methods (e.g. localization procedures), our approach appears less technical and provides short proofs. Besides, we are even able to deal with some classes of equations on rough domains. For instance, we can extend the well-known (and in general sharp) range for $p$ such that the (weak) Dirichlet Laplacian admits an ${\mathcal{H}}^\infty$-calculus on $L^p(\Omega)$, from $(3+\varepsilon)'<p<3+\varepsilon$ to $(4+\varepsilon)'<p<4+\varepsilon$ for three-dimensional bounded or unbounded Lipschitz cylinders $\Omega$. Our approach even admits mixed Dirichlet Neumann boundary conditions in this situation.