Revista Matemática Iberoamericana

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Published online first: 2019-12-17
DOI: 10.4171/rmi/1136

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

The Anh Bui[1], Piero D'Ancona[2] and Fabio Nicola[3]

(1) Macquarie University, Sydney, Australia
(2) Università di Roma La Sapienza, Italy
(3) Politecnico di Torino, Italy

We prove an $L^{p}$ estimate $$\| e^{-itL} \varphi(L)f \|_{p} \lesssim (1+|t|)^s \|f\|_p, \quad t\in \mathbb{R}, \quad s=n\Big|\frac{1}{2}-\frac{1}{p}\Big|$$ 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 $$\|e^{-itL} \varphi(\theta L) f\|_{p} \lesssim (1+\theta^{-1}|t|)^s \|f\|_p, \quad \theta > 0, \quad t\in \mathbb{R},$$ with uniformity also for $\theta$ varying in bounded subsets of $(0,+\infty)$. For nonnegative operators uniformity holds for all $\theta > 0$.

Keywords: Schrödinger group, metric measure spaces, doubling measure, spectral multipliers, heat kernels

Bui The Anh, D'Ancona Piero, Nicola Fabio: Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type. Rev. Mat. Iberoam. Electronically published on December 17, 2019. doi: 10.4171/rmi/1136 (to appear in print)