Weak type $(p,p)$ bounds for Schr\"odinger groups via generalized Gaussian estimates

TL;DR

This paper establishes weak type bounds for Schr"odinger groups at endpoint cases using generalized Gaussian estimates, extending known results to rough potentials and elliptic operators with less regular coefficients.

Contribution

It proves endpoint weak type $(p_0,p_0)$ bounds for operators $(I+L)^{-s_0} e^{itL}$ under generalized Gaussian estimates, covering more general operators with rough potentials.

Findings

Operators are of weak type $(p_0,p_0)$ with bounds depending on $t$

Results apply to Schr"odinger operators with rough potentials

Extends bounds to elliptic operators with measurable coefficients

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 $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. It is known that the operator $(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s\geq n|{1/ 2}-{1/p}| $ and $ p\in (p_0, p_0')$ (see for example, \cite{Blunck2, BDN, CCO, CDLY, DN, Mi1}). In this paper we study the endpoint case $p=p_0$ and show that for $s_0= n\big|{1\over 2}-{1\over p_0}\big|$, the operator $(I+L)^{-{s_0}}e^{itL} $ is of weak type $(p_{0},p_{0})$, that is, there is a constant $C>0$, independent of $t$ and $f$ so that \begin{eqnarray*} \mu\left(\left\{x: \big|(I+L)^{-s_0}e^{itL} f(x)\big|>\alpha \right\} \right)\leq C (1+|t|)^{n(1 - {p_0\over 2}) } \left( {\|f\|_{p_0} \over \alpha} \right)^{p_0} , \ \ \ t\in{\mathbb R} \end{eqnarray*} for $\alpha>0$ when $\mu(X)=\infty$, and $\alpha>\big(\|f\|_{p_{0}}/\mu(X) \big)^{p_{0}}$ when $\mu(X)<\infty$. Our results can be applied to Schr\"odinger operators with rough potentials and %second order elliptic operators with rough lower order terms, or higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds.