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Abstract
Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type with a dimension n. Suppose that the heat operator e- t L satisfies the generalized Gaussian (p0,p′0)-estimates of order m for some 1 ≤ p0< 2. In this paper we prove sharp endpoint Lp-Sobolev bound for the Schrödinger group eitL, that is for every p∈(p0,p′0) there exists a constant C= C(n, p) > 0 independent of t such that
∥(I+L)-seitLf∥p≤C(1+|t|)s‖f‖p, t∈R, s≥n|1/2-1/p|.
As a consequence, the above estimate holds for all 1 < p< ∞ when the heat kernel of L satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces Rn. We also give an application to obtain an endpoint estimate for Lp-boundedness of the Riesz means of the solutions of the Schrödinger equations.
Original language | English |
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Pages (from-to) | 667-702 |
Number of pages | 36 |
Journal | Mathematische Annalen |
Volume | 378 |
Issue number | 1-2 |
DOIs | |
Publication status | Published - Oct 2020 |
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Harmonic analysis and dispersive partial differential equations
Li, J., Guo, Z., Kenig, C. & Nakanishi, K.
31/01/17 → …
Project: Research
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Harmonic analysis: function spaces and partial differential equations
Duong, X., Hofmann, S., Ouhabaz, E. M. & Wick, B.
11/02/19 → 10/02/22
Project: Other