2021/03/18 by Akman, Murat, Hofmann, Steve, Martell, José María +1
#31B05 #35J08 #35J25 #42B25 #42B37 #42B99 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2103.10046
Let Ω⊂ℝn+1, n≥ 2, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider L0 u=-div(A0∇ u), Lu=-div(A∇ u), two real (non-necessarily symmetric) uniformly elliptic operators in Ω, and write ωL0, ωL for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ωL satisfies an A_∞-condition or a RHq-condition with respect to ωL0. In this paper we are interested in obtaining square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg-Jerison-Kenig and are fundamental for the proof of the perturbation results in arXiv:1901.08261.