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.
Sharp Lp estimates for Schrödinger groups on spaces of homogeneous type / Bui, T. A.; D'Ancona, P.; Nicola, F.. - In: REVISTA MATEMATICA IBEROAMERICANA. - ISSN 0213-2230. - 36:2(2020), pp. 455-484. [10.4171/rmi/1136]
Sharp Lp estimates for Schrödinger groups on spaces of homogeneous type
D'Ancona P.
;
2020
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.File | Dimensione | Formato | |
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