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

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