AI 中文总结
本文研究了Bergman投影在平面域上的弱型估计，通过共形映射分析边界几何对投影正则性的影响，并给出了弱型(1,1)和(p,p)界的条件及应用。
AI 中文摘要
我们通过共形映射\(\psi\colon\mathbb{D}\rightarrow\Omega\)，研究单连通区域\(\Omega\subset\mathbb{C}\)的伯格曼投影\(\Pi_{\Omega}\)的弱型正则性与\(\Omega\)边界几何之间的关系。证明当\(|\psi'|\)属于Bekollé - Bonami类\(B_1\)时，\(\Pi_{\Omega}\)是弱型\((1,1)\)；给出\(1\leq p<\infty\)时\(\Pi_{\Omega}\)的弱型\((p,p)\)界的更一般必要条件；建立\(p>1\)时弱型界的强化充分条件。结果源于对\(\Pi_{\mathbb{D}}\)的混合加权弱型不等式的重新表述，并给出了一些应用。
英文摘要
We investigate the relationship between the weak-type regularity of the Bergman projection, $Π_Ω$, of a simply connected domain $Ω\subset \mathbb{C}$ and the boundary geometry of $Ω$ in terms of a conformal map $ψ\colon\mathbb{D}\rightarrowΩ$. We show that $Π_Ω$ is of weak-type $(1,1)$ whenever $|ψ'|$ is in the Bekollé-Bonami class $B_1$, give a more general necessary condition for the weak-type $(p,p)$ bounds of $Π_Ω$ when $1\leq p<\infty$, and establish sharpened sufficient conditions for the weak-type bounds when $p>1$. Our results follow from a reformulation in terms of mixed-weighted weak-type inequalities for $Π_{\mathbb{D}}$. We provide several applications.
