## Abstract

We prove an L^{p} 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 L^{p0} → L^{p'0} smoothing estimate and a mild L^{2} → L^{2} 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 language | English |
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Pages (from-to) | 455-484 |

Number of pages | 30 |

Journal | Revista Matematica Iberoamericana |

Volume | 36 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

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