BMO Domains and Quasicircles Associated with Zygmund Functions

We investigate graph domains and graph curves from the viewpoints of BMOA, Zygmund regularity, and asymptotic geometry. First, we prove that every graph domain is a BMO domain, which provides a proof of a statement of Astala and Zinsmeister. We then use a Weierstrass-type function to construct a Bishop-Jones quasicircle that is asymptotically conformal but not asymptotically smooth. This example yields the strict inclusions \[ VMOA( H) BMOA( H)∩ B_0( H), SS( R) SQS( R)( R). \] Finally, by a localized lacunary construction, we produce a compactly supported locally absolutely continuous function f with f'( R)( R) whose graph is an asymptotically conformal chord-arc curve that is not asymptotically smooth.