Conformal welding of chord-arc curves

Abstract

We study the relation between the geometric properties of a chord-arc curve and its conformal welding. Let $h$ be the conformal welding of a closed quasicircle $\Gamma$. By Jones's theorem, the pull-back operator $C_h u=u\circ h$ is bounded on BMO if and only if $h$ corresponds to the welding of a Bishop-Jones quasicircle, equivalently, $h$ is strongly quasisymmetric. Let $A_h$ denote the analytic projection of $C_h$. We prove that $A_h$ is a bounded isomorphism on BMOA if and only if $\Gamma$ is a chord-arc curve. More strongly, the same characterization holds if invertibility is replaced by Fredholmness. This gives an intrinsic conformal-welding characterization of chord-arc curves and a complete geometric answer to the invertibility problem posed by Semmes in the 1980s. Furthermore, we establish an exact correspondence between the inverse of $A_h$ and the classical Faber integral operator, showing that for a rectifiable quasicircle, the Faber operator is a bounded isomorphism on BMOA if and only if the curve satisfies the chord-arc condition.