2026/07/16 by Katsuhiko Matsuzaki, Fei Tao
#math.CV
Let f\colon ℝ→ℝ be a continuous function. We define the graph welding associated with f as the homeomorphism φ= G-1∘ F\colon ℝ→ℝ. Here, F(x)=x+if(x) parametrizes the graph of f, and G is a conformal mapping from the upper half-plane ℍ onto one of the two domains bounded by the graph of f, admitting a continuous extension to ℝ. In this paper, we investigate how the regularity of the graph function f influences the analytic properties of the associated graph welding φ. In particular, under certain assumptions that f within Zygmund, BMO, 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 φ. These results clarify the interplay between the regularity of graph functions, the geometry of graph curves, and the boundary behavior of conformal mappings.