2024/11/21 by Utsav Dewan, Dewan, Utsav
Mathematics · #43A85 #43A90 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Primary 35J10 #Secondary 22E30 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2411.14020
openalex publication_date 2024/11/21 · openalex created_date 2024/11/24 · openalex updated_date 2026/07/28
One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition f of the Schrödinger equation given by \begincases i(∂ u)/(∂ t) =Δu , (x,t) ∈ ℝn × ℝ \newline u(0,⋅)=f , on ℝn , \endcases in terms of the index β such that f belongs to the inhomogeneous Sobolev space Hβ(ℝn), so that the solution of the Schrödinger operator u converges pointwise to f, limt → 0+ u(x,t)=f(x), almost everywhere. Recently, the author considered the Carleson's problem for the Schrödinger equation with radial initial data on Damek-Ricci spaces and obtained the sharp bound up to the endpoint β≥ 1/4. Interpreting the above as convergence along vertical lines, in this article, we consider the problem of pointwise convergence via more general approach paths. By constructing a counter-example on the 3-dimensional Real Hyperbolic space, we show that the solutions of the Schrödinger equation, unlike Harmonic functions or solutions of the Heat equation, do not admit any natural wide approach region. We then study their pointwise convergence properties on Damek-Ricci spaces along general curves that satisfy certain Hölder conditions and bilipschitz conditions in the distance from the identity and again obtain the sharp bound up to the endpoint β≥ 1/4. Certain Euclidean analogues are also obtained.