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arXiv · 2606.16798

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

Xiaosheng Lin·Dachun Yang·Sibei Yang·Wen Yuan·Yangyang Zhang
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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.

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Xiaosheng Lin, Dachun Yang, Sibei Yang, Wen Yuan, Yangyang Zhang. 2026-06-15. A Counterexample to the Necessity of the Vanishing Carleson Condition for VMO Poisson Kernels. https://arxiv.org/abs/2606.16798

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