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In the 80’s Dahlberg asked two questions regarding the `$L^p$ – solvability’ of elliptic equations with variable coefficients. Dahlberg’s first question was whether $L^p$ solvability was maintained under `Carleson-perturbations’ of the coefficients. This question was answered by Fefferman, Kenig and Pipher [FKP], where they also introduced new characterizations of $A_\infty$, reverse-Hölder and $A_p$ weights. These characterizations were used to create a counterexample to show their theorem was sharp.
Dahlberg’s second question was whether a Carleson gradient/oscillation condition (the `DKP condition’) was enough to imply $L^p$ solvability for some p > 1. This was answered by Kenig and Pipher [KP] and refined by Dindos, Petermichl and Pipher [DPP] (in the `small constant’ case). These $L^p$ solvability results can be interpreted in terms of a reverse Hölder condition for the elliptic kernel and therefore connected with the $A_\infty$ condition. In this talk, we discuss L^p solvability for a class of coefficients that satisfies a `weak DKP condition’. In particular, we connect the (weak) DKP condition to the characterization of $A_\infty$ in [FKP]. This allows us to treat the `large’, `small’ and ‘vanishing’ (weak) DKP conditions simultaneously and independently from the works [KP] and [DPP].
This is joint work with my co-authors Egert, Saari, Toro and Zhao. A proof of the main estimate will be sketched, but technical details will be avoided.