2024/07/09 by Manli Song, Song, Manli, Jinggang Tan +1 · 2 citations
Mathematics · #22E25 #33C45 #35B40 #35H20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2407.06899
openalex publication_date 2024/07/09 · openalex created_date 2024/07/11 · openalex updated_date 2026/07/28
Let L be the sub-Laplacian on H-type groups and ϕ: ℝ+ → ℝ be a smooth function. The primary objective of the paper is to study the decay estimate for a class of dispersive semigroup given by eitϕ(L). Inspired by earlier work of Guo-Peng-Wang \citeGPW2008 in the Euclidean space and Song-Yang \citeSY2023 on the Heisenberg group, we overcome the difficulty arising from the non-homogeneousness of ϕ by frequency localization, which is based on the non-commutative Fourier transform on H-type groups, the properties of the Laguerre functions and Bessel functions, and the stationary phase theorem. Finally, as applications, we derive the new Strichartz inequalities for the solutions of some specific equations, such as the fractional Schrödinger equation, the fourth-order Schrödinger equation, the beam equation and the Klein-Gordon equation, which corresponds to ϕ(r)=rα, r2+r,√(1+r2),√(1+r), respectively. Moreover, we also prove that the time decay is sharp in these cases.