Graph Weldings Associated with Functions in Zygmund, $\mathrm{BMO}$, $\mathrm{VMO}$, and $H^{1/2}$

Abstract

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. We define the graph welding associated with $f$ as the homeomorphism \[ φ= G^{-1}\circ F\colon \mathbb{R}\to\mathbb{R}. \] Here, $F(x)=x+if(x)$ parametrizes the graph of $f$, and $G$ is a conformal mapping from the upper half-plane $\mathbb{H}$ onto one of the two domains bounded by the graph of $f$, admitting a continuous extension to $\mathbb{R}$. In this paper, we investigate how the regularity of the graph function $f$ influences the analytic properties of the associated graph welding $\phi$. In particular, under certain assumptions that $f$ within Zygmund, $\mathrm{BMO}$, $\mathrm{VMO}$, and Hardy spaces, we establish results on absolute continuity, quasisymmetry, symmetry, strong quasisymmetry, strong symmetry, and the Weil--Petersson property of the associated graph welding $\phi$. These results clarify the interplay between the regularity of graph functions, the geometry of graph curves, and the boundary behavior of conformal mappings.