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.
Categories: math.AP