Beurling-Ahlfors extension by heat kernel, ${\mathrm{A}}_{\mathrm{\infty }}$-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 ${\mathrm{A}}_{\mathrm{\infty }}$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.