Conformal welding of chord-arc curves

Abstract: We study the relationship between the geometric properties of a chord-arc curve and its conformal welding. Let $h$ be the conformal welding of a closed Jordan curve $Γ$. By Jones's theorem, the pull-back operator $C_h$ is bounded on BMO if and only if $h$ corresponds to the welding of a Bishop-Jones quasi-circle. Letting $A_h$ denote the analytic projection of $C_h$, we prove that $A_h$ is a bounded isomorphism on BMOA if and only if $Γ$ is a chord-arc curve. This provides a complete conformal welding characterization of chord-arc curves and resolves an open problem proposed 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 curve, the Faber operator is a bounded isomorphism on BMOA if and only if the curve satisfies the chord-arc condition.