Weak type $(p,p)$ bounds for Schrödinger groups via generalized Gaussian estimates

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 $(p_0,p_0^{\prime })$-estimates of order $m$ for some $1\le p_0<2$. It is known that the operator $(I+L{)}^{-s}{e}^{itL}$ is bounded on $L^p(X)$ for $s\ge n|1/2-1/p|$ and $p\in (p_0,p_0^{\prime })$ (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 $s_0=n|\frac{1}{2}-\frac{1}{p_0}|$, 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*} μ\left(\left\{x: \big|(I+L)^{-s_0}e^{itL} f(x)\big|>α\right\} \right)\leq C (1+|t|)^{n(1 - {p_0\over 2}) } \left( {\|f\|_{p_0} \over α} \right)^{p_0} , \ \ \ t\in{\mathbb R} \end{eqnarray*} for $\alpha >0$ when $\mu (X)=\infty$, and $\alpha >(\Vert f{\Vert }_{p_0}/\mu (X){)}^{p_0}$ when $\mu (X)<\infty$. Our results can be applied to Schrödinger 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.

17 pages