2024/11/26 by Dumont, Arnaud, Andrew J. Morris, Morris, Andrew J.
Mathematics · #35J25 #35J25 (Primary) 35J10 #42B37 #47A60 (Secondary) #47D06 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2411.17563
openalex publication_date 2024/11/26 · openalex created_date 2024/12/05 · openalex updated_date 2026/07/28
We obtain Riesz transform bounds and characterise operator-adapted Hardy spaces to solve boundary value problems for singular Schrödinger equations -div(A∇ u)+aVu=0 in the upper half-space ℝ1+n+ with boundary dimension n≥ 3. The coefficients (A,a,V) are assumed to be independent of the transversal direction to the boundary, and consist of a complex-elliptic pair (A,a) that is bounded and measurable with a certain block structure, and a non-negative singular potential V in the reverse Hölder class RHq(ℝn) for q≥ max\(n)/(2),2\. This block structure is significant because it allows for coefficients that are not symmetric but for which L2(ℝn)-solvability persists due to recently obtained Kato square root type estimates. We find extrapolation intervals for exponents p around 2 on which the Dirichlet problem is well-posed for boundary data in Lp(ℝn), and the associated Regularity problem is well-posed for boundary data in Sobolev spaces V1,p(ℝn) that are adapted to the potential V, when p>1. The well-posedness of these Dirichlet problems and related estimates then allow us to solve the corresponding Neumann problem with boundary data in Lp. The results permit boundary data in the Dziubanski--Zienkiewicz Hardy space H1V(ℝn) and adapted Hardy--Sobolev spaces H1,pV(ℝn) when p≤ 1. We also obtain comparability of square functions and nontangential maximal functions for the solutions with their boundary data.