Sharp Lp estimates for Schrödinger groups on spaces of homogeneous type

The Anh Bui, Piero D'Ancona, Fabio Nicola

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We prove an Lp estimate (Equation Presented) for the Schrödinger group generated by a semibounded, self-adjoint operator L on a metric measure space X of homogeneous type (where n is the doubling dimension of X). The assumptions on L are a mild Lp0 → Lp'0 smoothing estimate and a mild L2 → L2 off-diagonal estimate for the corresponding heat kernel e-tL. The estimate is uniform for φ varying in bounded sets of S(R), or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on L, that the estimate extends to (Equation Presented) with uniformity also for θ varying in bounded subsets of (0,+∞). For nonnegative operators uniformity holds for all θ > 0.

Original languageEnglish
Pages (from-to)455-484
Number of pages30
JournalRevista Matematica Iberoamericana
Volume36
Issue number2
DOIs
Publication statusPublished - 2020

Keywords

  • Schrödinger group
  • metric measure spaces
  • doubling measure
  • spectral multipliers
  • heat kernels

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