The Schrödinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds

Abstract

Let L be a non-negative self-adjoint operator acting on L 2 (X) where X is a space of homogeneous type with a dimension n. In this paper, we study sharp endpoint L p -Sobolev estimates for the solution of the initial value problem for the Schrödinger equation i∂ t u + Lu = 0 and show that for all f ∈ L p (X), 1 < p < ∞,where the semigroup e −tL generated by L satisfies a Poisson type upper bound. This extends the previous result in [8] in which the semigroup e −tL generated by L satisfies the exponential decay.

On complex-time heat kernels of fractional Schrödinger operators via Phragmén-Lindelöf principle

“…Complex-time heat kernel estimates like (1.2) are of paramount importance in many problems in harmonic analysis and partial differential equations, e.g., in proving L p boundedness of spectral multipliers and convergence of Riesz means, and investigating maximal regularity properties of the Schrödinger evolution e itH for operators H whose heat kernels satisfy (sub-)gaussian estimates, see, e.g., [4,9,7,5,6,8,12,13,15,14,16,17,18,23,24,29,42].…”