Compactness characterization for the anti-analytic projection

Determine whether the projected composition operator \(\bar{A}_h=P_-C_h\big|_{\mathrm{BMOA}}\) is compact if and only if the welded Jordan curve \(\Gamma\) is asymptotically smooth.

Background

For a strongly quasisymmetric welding $h$, the pull-back operator $C_h$ on BMO decomposes into analytic and anti-analytic projected composition operators. The paper proves a chord-arc characterization in terms of invertibility of the analytic projection $A_h$, while the related compactness behavior of $\bar A_h$ is connected to asymptotically smooth curves, equivalently to the VMO condition on $\log h'$. The authors report this as a conjecture of Fan, Hu, and Shen and state that it will be treated separately, so it remains unresolved in the present paper.