Beurling–Ahlfors extension by heat kernel, A∞‐weights for VMO, and vanishing Carleson measures

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 A ∞ -weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.