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 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