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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.

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