Weak Type (1,1) Bounds for Schrödinger Groups

Abstract

Let L be a nonnegative self-adjoint operator acting on ${\mathit{L}}^2(\mathit{X})$, where X is a space of homogeneous type of dimension n. Suppose that the heat kernel of L satisfies a Gaussian upper bound. It is known that the operator ${(\mathit{I}+\mathit{L})}^{-\mathit{s}}{\mathit{e}}^{\mathit{i}\mathit{t}\mathit{L}}$ is bounded on ${\mathit{L}}^{\mathit{p}}(\mathit{X})$ for $\mathit{s}>\mathit{n}|1/2-1/\mathit{p}|$ and $\mathit{p}\in (1,\infty )$ (see, e.g., [7, 22, 33]). The index $\mathit{s}=\mathit{n}|1/2-1/\mathit{p}|$ was only obtained recently in [9, 10], and this range of s is sharp since it is precisely the range known in the case where L is the Laplace operator Δ on $\mathit{X}={\mathbb{R}}^{\mathit{n}}$ [30]. In this paper, we establish that for $\mathit{p}=1$, the operator ${(1+\mathit{L})}^{-\mathit{n}/2}{\mathit{e}}^{\mathit{i}\mathit{t}\mathit{L}}$ is of weak type $(1,1)$, that is, there is a constant C, independent of t and f, such that

(for $\mathit{\lambda }>0$ when $\mathit{\mu }(\mathit{X})=\infty$ and $\mathit{\lambda }>\mathit{\mu }{(\mathit{X})}^{-1}\Vert \mathit{f}{\Vert }_{{\mathit{L}}^1(\mathit{X})}$ when $\mathit{\mu }(\mathit{X})<\infty$). Moreover, we also show that the index $\mathit{n}/2$ is sharp when L is the Laplacian on ${\mathbb{R}}^{\mathit{n}}$ by providing an example.

Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.