  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.

Complex Variables;Analysis of PDEs