Beurling-Ahlfors extension by heat kernel, ${\rm A}_\infty$-weights for VMO, and vanishing Carleson measures

TL;DR

This paper introduces a heat kernel-based Beurling-Ahlfors extension for quasisymmetric homeomorphisms, showing that certain weights induce vanishing Carleson measures, linking complex analysis and harmonic analysis.

Contribution

It presents a novel heat kernel convolution method for Beurling-Ahlfors extension and establishes a connection between ${ m A}_ Infty$-weights, VMO, and vanishing Carleson measures.

Findings

The heat kernel extension preserves strong symmetry properties.

${ m A}_ Infty$-weights' logarithm in VMO implies vanishing Carleson measures.

New link between quasiconformal extensions and harmonic analysis measures.

Abstract

We investigate a variant of the Beurling-Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (i.e. its derivative is an ${\rm A}_\infty$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.