Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces

2025/06/01 by Utsav Dewan, Dewan, Utsav
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.00881

Abstract

We study the Carleson's problem on Damek-Ricci spaces S for dispersive equations: \begincases i(∂ u)/(∂ t) +Ψ(√(-L) )u=0 , (x,t) ∈ S × ℝ ,
u(0,⋅)=f , on S , \endcases where L= Δ, the Laplace-Beltrami operator or Δ, the shifted Laplace-Beltrami operator, so that the corresponding phase function ψ satisfies for some a ∈ (0,1), the large frequency asymptotic: ψ(λ)=λ^a + O(1) , λ≫ 1 . For almost everywhere pointwise convergence of the solution u to its radial initial data f, we obtain the almost sharp regularity threshold β>a/4. This result is new even for ℝⁿ and in the special case of the fractional Schrödinger equations, generalizes classical Euclidean results of Walther.

Citations

• Maximal estimates and pointwise convergence for solutions of certain dispersive equations with radial initial data on Damek-Ricci spaces
• Mapping properties of the Schrödinger maximal function on Damek--Ricci spaces
• The wave equation on Damek-Ricci spaces
• Interpolation of linear operators

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