Beurling-Ahlfors extension by heat kernel, ${rm A}_infty$-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 (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.