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 (that is, its derivative is an ${\mathrm{A}}_{\infty }$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.

中文翻译：

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热核的 Beurling-Ahlfors 扩展、VMO 的 A∞ 权重和消失的 Carleson 测度

我们研究了由热核的卷积给出的实线的拟对称同胚的 Beurling-Ahlfors 扩展的变体，并证明了强对称同胚的这种拟共形扩展的复膨胀（即它的导数）是一个 ${\mathrm{一种}}_{\infty }$-weight 的对数在 VMO 中）导致在上半平面上消失的卡尔森测度。