Abstract
Let $L$ be a non-negative self-adjoint 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 $i \partial_{t} u + L u = 0$ and show that for all $f \in L^{p}(X)$, $1 < p < \infty$, $\| e^{itL} (I+L)^{-{\sigma n}} f\|_{p} \leq C(1 + |t|)^{\sigma n} \| f \|_{p}$, $t \in \mathbb{R}$, $\sigma \geq |1/2-1/p|$, where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound.
Funding Statement
The first author was supported by NNSF of China 11501583, Guangdong Natural Science Foundation 2016A030313351. The second author is supported by the Australian Research Council (ARC) through the research grants DP190100970. The third author was supported by International Program for Ph.D. Candidates from Sun Yat-Sen University. The fourth author is supported by the Australian Research Council (ARC) through the research grant DP170101060 and by Macquarie University Research Seeding Grant. The fifth author was supported by the NNSF of China, Grant No. 11871480, and by the Australian Research Council (ARC) through the research grants DP190100970.
Citation
Peng CHEN. Xuan Thinh DUONG. Zhijie FAN. Ji LI. Lixin YAN. "The Schrödinger equation in $L^{p}$ spaces for operators with heat kernel satisfying Poisson type bounds." J. Math. Soc. Japan 74 (1) 285 - 331, January, 2022. https://doi.org/10.2969/jmsj/85278527
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