A Counterexample to the Necessity of the Vanishing Carleson Condition for VMO Poisson Kernels

Abstract

In [Problem 3.2.23, CBMS Regional Conference Series in Mathematics 83, 1994], Kenig asked whether the vanishing Carleson condition is the necessary and sufficient for the logarithm of the Poisson kernel of a perturbation of the Laplacian on the unit ball in $\mathbb{R}^n$ belonging to the space VMO. The sufficiency was proved by Escauriaza [Israel J. Math. 1996] and extended by Milakis, Pipher, and Toro [Contemp. Math. 2014] to more general setting. In this article, using the technique of bi-Lipschitz mappings, we construct a counterexample to show that the vanishing Carleson condition is not necessary and hence give a negative answer to the aforementioned problem.