2024/01/12 by Damian Dąbrowski, Xavier Tolsa, Dąbrowski, Damian +1 · 1 citation
Computer Science · Mathematics · #28A75 #42B20 #49Q15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2402.08615
openalex publication_date 2024/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
In this work we provide a geometric characterization of the measures μ in \mathbb Rn+1 with polynomial upper growth of degree n such that the n-dimensional Riesz transform Rμ(x) = ∫ \fracx-y|x-y|n+1 dμ(y) belongs to L2(μ). More precisely, it is shown that ‖Rμ‖L2(μ)2 + ‖μ‖≈ ∫ ∫0^∞ β2,μ(x,r)2 (μ(B(x,r)))/(rn) \fracdrr dμ(x) + ‖μ‖, where βμ,2(x,r)2 = infL \frac1rn∫B(x,r) (\fracdist(y,L)r)2 dμ(y), with the infimum taken over all affine n-planes L⊂\mathbb Rn+1. As a corollary, we obtain a characterization of the removable sets for Lipschitz harmonic functions in terms of a metric-geometric potential and we deduce that the class of removable sets for Lipschitz harmonic functions is invariant by bilipschitz mappings.