# Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type

### The Anh Bui

Macquarie University, Sydney, Australia### Piero D'Ancona

Università di Roma La Sapienza, Italy### Fabio Nicola

Politecnico di Torino, Italy

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## Abstract

We prove an $L^{p}$ estimate

for the Schrödinger group generated by a semibounded, self-adjoint operator $L$ on a metric measure space $\mathcal{X}$ of homogeneous type (where $n$ is the doubling dimension of $\mathcal{X}$). The assumptions on $L$ are a mild $L^{p_{0}}\to L^{p_{0}'}$ smoothing estimate and a mild $L^{2}\to L^{2}$ off-diagonal estimate for the corresponding heat kernel $e^{-tL}$. The estimate is uniform for $\varphi$ varying in bounded sets of $\mathscr{S}(\mathbb{R})$,or more generally of a suitable weighted Sobolev space.

We also prove, under slightly stronger assumptions on $L$, that the estimate extends to

with uniformity also for $\theta$ varying in bounded subsets of $(0,+\infty)$. For nonnegative operators uniformity holds for all $\theta > 0$.

## Cite this article

The Anh Bui, Piero D'Ancona, Fabio Nicola, Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type. Rev. Mat. Iberoam. 36 (2019), no. 2, pp. 455–484

DOI 10.4171/RMI/1136