The Schr\"odinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds

TL;DR

This paper establishes sharp endpoint $L^p$-Sobolev estimates for the Schr"odinger equation with operators whose heat kernels satisfy Poisson type bounds, extending previous results for exponential decay cases.

Contribution

It extends previous Schr"odinger equation estimates to operators with Poisson type heat kernel bounds, broadening the class of operators for which these estimates hold.

Findings

Proves $L^p$-Sobolev estimates for Schr"odinger solutions with Poisson bounds.

Shows the estimates depend on the parameter $ig|{1/2}-{1/p}ig|$ and time $t$.

Generalizes earlier results from exponential decay to Poisson 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 Schr\"odinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \begin{eqnarray*} \left\| e^{itL} (I+L)^{-{\sigma n}} f\right\|_{p} \leq C(1+|t|)^{\sigma n} \|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ \sigma\geq \big|{1\over 2}-{1\over p}\big|, \end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound. This extends the previous result in \cite{CDLY1} in which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.