二维变量有理函数的边界局部积分性
Boundary local integrability of rational functions in two variables
AI总结：
研究二维变量有理函数在原点附近局部积分性，基于Bickel-Pascoe-Sola的工作，给出局部L^p空间的必要充分条件，并描述所有满足条件的分子Q。
摘要：
受研究二维变量有理函数在域上分析边界奇点的启发，我们探讨了在双上半平面中分母不消失但原点处有孤立零点的有理函数在R^2上的局部积分性。基于Bickel-Pascoe-Sola的工作，我们给出了属于局部L^p(R^2)空间的必要充分检验条件，并完整描述了所有使得Q/P属于给定L^p空间的分子Q。作为应用，我们证明了在双圆盘上每个多项式有理函数的偏导数都属于L^1。此外，我们给出了一个猜想的新证明，该猜想由Bickel-Knese-Pascoe-Sola提出并由Kollár完成，用于表征Q/P局部有界的理想。这项工作的一个更大收获是，我们所采用的稳定多项式局部模型是一种灵活的工具，可能对其他关于稳定多项式的局部问题有所帮助。
英文摘要：
Motivated by studying boundary singularities of rational functions in two variables that are analytic on a domain, we investigate local integrability on $\mathbb{R}^2$ near $(0,0)$ of rational functions with denominator non-vanishing in the bi-upper half-plane but with an isolated zero (with respect to $\mathbb{R}^2$) at the origin. Building on work of Bickel-Pascoe-Sola, we give a necessary and sufficient test for membership in a local $L^{p}(\mathbb{R}^2)$ space and we give a complete description of all numerators $Q$ such that $Q/P$ is locally in a given $L^{p}$ space. As applications, we prove that every bounded rational function on the bidisk has partial derivatives belonging to $L^1$ on the two-torus. In addition, we give a new proof of a conjecture, started in Bickel-Knese-Pascoe-Sola and completed by Kollár, characterizing the ideal of $Q$ such that $Q/P$ is locally bounded. A larger takeaway from this work is that a local model for stable polynomials we employ is a flexible tool and may be of use for other local questions about stable polynomials.
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