$\mathrm{BMO}$ Domains and Quasicircles Associated with Zygmund Functions

We investigate graph domains and graph curves from the viewpoints of $\mathrm{BMOA}$, Zygmund regularity, and asymptotic geometry. First, we prove that every graph domain is a $\mathrm{BMO}$ 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 \[ \mathrm{VMOA}(\mathbb H) \subsetneq \mathrm{BMOA}(\mathbb H)\cap B_0(\mathbb H), \qquad \mathrm{SS}(\mathbb R) \subsetneq \mathrm{SQS}(\mathbb R)\cap\mathrm{S}(\mathbb R). \] Finally, by a localized lacunary construction, we produce a compactly supported locally absolutely continuous function $f$ with $f'\in\mathrm{BMO}(\mathbb R)\setminus\mathrm{VMO}(\mathbb R)$ whose graph is an asymptotically conformal chord-arc curve that is not asymptotically smooth.