Lp and operator norm estimates for the complex time heat operator on homogeneous trees

Let X be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator Hζ induced by the natural Laplace operator on X. We prove comparable upper and lower bounds for the Lp norms of its convolution kernel hζ and derive precise estimates for the Lp-Lr operator norms of Hζ for ζ belonging to the half plane Re ζ ≥ 0. In particular, when ζis purely imaginary, our results yield a description of the mapping properties of the Schrödinger semigroup on X. ©1998 American Mathematical Society.