On smoothing estimates for Schrödinger equations on product spaces \mathbbT^m× ℝⁿ

2023/01/13 by Chen, Xianghong, Guo, Zihua, Shen, Minxing +1
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.05450

Abstract

Let Δ_\mathbbT^m× ℝⁿ denote the Laplace-Beltrami operator on the product spaces \mathbbT^m× ℝⁿ. In this article we show that ‖e^itΔ_\mathbbT^m× ℝⁿf‖_L^p(\mathbbT^m× ℝⁿ× [0,1]) ≤ C ‖f‖_W^α,p(\mathbbT^m×ℝⁿ) holds if p≥ 2(m+n+2)/(m+n) and α> (m+2n)(1/2-1/p)-2/p. Furthermore, we apply the ℓ²-decoupling inequalities to establish local L^p-smoothing estimates for the Schrödinger operator e^itΔ_\mathbbT^m×ℝⁿ in modulation spaces M_p,q^α(\mathbbT^m×ℝⁿ): ‖e^itΔ_\mathbbT^m×ℝⁿf‖_L^p(\mathbbT^m×ℝⁿ× [0,1])≤ C ‖f‖_M_p,q^α(\mathbbT^m×ℝⁿ) for some range of α and p, q. The smoothing estimates in L^p-Sobolev and modulation spaces are sharp up to the endpoint regularity, in a certain range of p and q.

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