AI 中文总结
研究连续函数\(f\)相关的图焊接\(\varphi\)，在\(f\)属于齐格蒙德、\(\mathrm{BMO}\)、\(\mathrm{VMO}\)和哈代空间某些假设下，探讨\(f\)正则性对\(\varphi\)解析性质的影响，建立了多种性质的相关结果，阐明了三者间相互作用。
AI 中文摘要
设\(f\colon \mathbb{R}\to\mathbb{R}\)为连续函数。定义与\(f\)相关的图焊接为同胚映射\(\varphi = G^{-1}\circ F\colon \mathbb{R}\to\mathbb{R}\)。其中\(F(x)=x + if(x)\)参数化\(f\)的图像，\(G\)是从上半平面\(\mathbb{H}\)到由\(f\)的图像界定的两个区域之一的共形映射，且能连续延拓到\(\mathbb{R}\)。本文研究图函数\(f\)的正则性如何影响相关图焊接\(\varphi\)的解析性质。特别是在\(f\)属于齐格蒙德、\(\mathrm{BMO}\)、\(\mathrm{VMO}\)和哈代空间的某些假设下，建立了关于相关图焊接\(\varphi\)的绝对连续性、拟对称性、对称性、强拟对称性、强对称性以及韦伊 - 彼得森性质的结果。这些结果阐明了图函数正则性、图曲线几何以及共形映射边界行为之间的相互作用。
英文摘要
Let $f\colon \mathbb{R}\to\mathbb{R}$ 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 $φ$. 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 $φ$. These results clarify the interplay between the regularity of graph functions, the geometry of graph curves, and the boundary behavior of conformal mappings.
