Sharp endpoint estimates for Schrödinger groups on Hardy spaces

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 kernel of $L$ satisfies the Davies-Gaffney estimates of order $m\ge 2$. Let $H_L^1(X)$ be the Hardy space associated with $L.$ In this paper we show sharp endpoint estimate for the Schrödinger group $e^{itL}$ associated with $L$ such that \begin{eqnarray*} \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ L^1(X)} + \left\| (I+L)^{-{n/2}}e^{itL} f\right\|_{ H^1_L(X)} \leq C(1+|t|)^{n/2}\|f\|_{H^1_L(X)}, \ \ \ t\in{\mathbb R} \end{eqnarray*} for some constant $C=C(n,m)>0$ independent of $t$. By a duality and interpolation argument, it gives a new proof of a recent result of \cite{CDLY} for { sharp} endpoint $L^p$-Sobolev bound for $e^{itL}$: $${\Vert (I+L{)}^{-s}{e}^{itL}f\Vert }_{L^p(X)}\le C(1+|t|{)}^s\Vert f{\Vert }_{L^p(X)},t\in \mathbb{R},s\ge n|\frac{1}{2}-\frac{1}{p}|$$ for every $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound, which extends the classical results due to Miyachi ) for the Laplacian on the Euclidean space ${\mathbb{R}}^n$.

References in corpus (1)

• Gaussian heat kernel upper bounds via Phragmén-Lindelöf theorem