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We prove an estimate
for the Schrödinger group generated by a semibounded, self-adjoint operator on a metric measure space of homogeneous type (where is the doubling dimension of ). The assumptions on are a mild smoothing estimate and a mild off-diagonal estimate for the corresponding heat kernel . The estimate is uniform for varying in bounded sets of ,or more generally of a suitable weighted Sobolev space.
We also prove, under slightly stronger assumptions on , that the estimate extends to
with uniformity also for varying in bounded subsets of . For nonnegative operators uniformity holds for all .
Cite this article
The Anh Bui, Piero D'Ancona, Fabio Nicola, Sharp estimates for Schrödinger groups on spaces of homogeneous type. Rev. Mat. Iberoam. 36 (2020), no. 2, pp. 455–484DOI 10.4171/RMI/1136