Absolute continuity of degenerate elliptic measure

Let Ω⊂Rn+1 be an open set whose boundary may be composed of pieces of different dimensions. Assume that Ω satisfies the quantitative openness and connectedness, and there exist doubling measures m on Ω and μ on ∂Ω with appropriate size conditions. Let Lu=−div(A∇u) be a real (not necessarily symmetric) degenerate elliptic operator in Ω. Write ω<inf>L</inf> for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) ω<inf>L</inf>∈A<inf>∞</inf>(μ), (ii) the Dirichlet problem for L is solvable in Lp(μ) for some p∈(1,∞), (iii) every bounded null solution of L satisfies Carleson measure estimates with respect to μ, (iv) the conical square function is controlled by the non-tangential maximal function in Lq(μ) for all q∈(0,∞) for any null solution of L, and (v) the Dirichlet problem for L is solvable in BMO(μ). On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of ω<inf>L</inf> with respect to μ in terms of local L2(μ) estimates of the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness μ-almost everywhere of the truncated conical square function for any bounded null solution of L. © 2024 The Author(s)