2023/10/21 by Francesco Di Plinio, Di Plinio, Francesco, A. Walton Green +3
Mathematics · #42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Primary: 30C62. Secondary: 42B20
paper · pdf · doi:10.48550/arxiv.2310.14089
openalex publication_date 2023/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We quantify the Sobolev space norm of the Beltrami resolvent (I- μB)-1, where \mathcal B is the Beurling-Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation μ in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in W1,p, p ≥ 2. Our proof strategy is then adapted to yield quantitative estimates for the resolvent (I-μ\mathcal BΩ)-1 of the Beltrami equation on a sufficiently regular domain Ω, with μ∈ W1,p(Ω). Here, \mathcal BΩ is the compression of \mathcal B to a domain Ω. Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón-Zygmund operators to domains, recently obtained by the authors, to extend the Astala-Iwaniec-Saksman technique to higher regularities.