Functional calculus on Venturi for Groups with Finite Propagation Speed

Abstract

Let $\mathcal{M}$ be a complete Riemannian manifold with Ricci curvature bounded below and Laplace operator $\hat{I}”$. The paper develops a functional calculus for the cosine family $\cos (t\sqrt{\hat{I}}”)$ which is associated with waves that travel at unit speed. If $f$ is holomorphic on a Venturi shaped region, and $z^{k}f(z)$ is bounded for some positive integer $k$, then $f(\sqrt{\hat{I}}”)$ defines a bounded linear operator on $L^p(\mathcal{M})$ for some $p>2$. For Jacobi hypergroups with invariant measure $m$ the generalized Fourier transform of $f\in L^1(m)$ gives $\hat{f}\in H^{\infty }(\hat{I}{\pounds }_{\ddot{I}}‰)$ for some strip $\hat{I}{\pounds }_{\ddot{I}}‰$. Hence one defines $\hat{f}(A)$ for operators $A$ in some Banach space that have a $H^{\infty }(\hat{I}{\pounds }_{\ddot{I}}‰)$ functional calculus. The paper introduces an operational calculus for the Mehler--Fock transform of order zero. By transference methods, one defines $\hat{f}(A)$ when $\hat{f}$ is a $s$-Marcinkiewicz multiplier and $e^{itA}$ is a strongly continuous operator group on a $L^p$ space for $|1/2-1/p|<1/s$.\par

28 pages. This manuscript will not be published. Many of the ideas are included in a more recent paper arXiv:1509.00133