2021/06/01 by Dąbrowski, Damian, Tolsa, Xavier
#42B20 (Primary) 28A75 #49Q15 (Secondary) #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.00303
In this work we obtain a geometric characterization of the measures μ in 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), under the assumption that μ satisfies the following Wolff energy estimate, for any ball B⊂ℝn+1: ∫B ∫0^∞ (\fracμ(B(x,r))rn-\frac38)2 \fracdrr dμ(x)≤ M (\fracμ(2B)r(B)n-\frac38)2 μ(2B). More precisely, we show that μ satisfies the following estimate: ‖Rμ‖L2(μ)2 + ‖μ‖≈ ∫ ∫0^∞ βμ,2(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⊂ℝn+1. In a companion paper which relies on the results obtained in this work it is shown that the same result holds without the above assumption regarding the Wolff energy of μ. This result has important consequences for the Painlevé problem for Lipschitz harmonic functions.