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Abstract
Let L be a nonnegative selfadjoint operator acting on L^{2}(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 (p_{0},p′_{0})estimates of order m for some 1 ≤ p0< 2. In this paper we prove sharp endpoint L^{p}Sobolev bound for the Schrödinger group e^{itL}, that is for every p∈(p_{0},p′_{0}) there exists a constant C= C(n, p) > 0 independent of t such that
∥(I+L)^{s}e^{itL}f∥_{p}≤C(1+t)^{s}‖f‖_{p}, t∈R, s≥n1/21/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 R^{n}. We also give an application to obtain an endpoint estimate for L^{p}boundedness of the Riesz means of the solutions of the Schrödinger equations.
Original language  English 

Pages (fromto)  667702 
Number of pages  36 
Journal  Mathematische Annalen 
Volume  378 
Issue number  12 
DOIs  
Publication status  Published  Oct 2020 
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Projects
 2 Active

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

Harmonic analysis and dispersive partial differential equations
Li, J., Guo, Z., Kenig, C. & Nakanishi, K.
31/01/17 → …
Project: Research