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.
We introduce the concept of a new kind of symmetric homeomorphism on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we show that the Bers Schwarzian derivative map is a holomorphic split submersion and endow a complex Banach manifold structure on the Teichmüller space of those generalized symmetric homeomorphisms.
Following Semmes [16] and Zinsmeister [20], we continue the study of Carleson measures and their invariance under pull-back and push-forward operators. We also study the analogous statements for vanishing Carleson measures. As an application, we show that some quotient space of the space of chord-arc curves has a natural complex structure.
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