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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. In this paper, we study sharp endpoint L^{p}Sobolev estimates for the solution of the initial value problem for the Schrödinger equation i∂tu+Lu = 0 and show that for all f ∈ L^{p}(X), 1 < p < ∞, ∥e^{itL}(I + L)^{−σn}f∥p ≤ C(1 + t)^{σn}∥f∥p, t ∈ R, σ ≥ 1/2 − 1/p, where the semigroup e^{−tL} generated by L satisfies a Poisson type upper bound.
Original language  English 

Pages (fromto)  285331 
Number of pages  47 
Journal  Journal of the Mathematical Society of Japan 
Volume  74 
Issue number  1 
DOIs  
Publication status  Published  Jan 2022 
Keywords
 sharp Lp estimate
 Schrödinger equation
 elliptic operator
 heat kernel
 space of homogeneous type
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Harmonic analysis and dispersive partial differential equations
Li, J., Guo, Z., Kenig, C. & Nakanishi, K.
31/01/17 → …
Project: Research

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