标题： 复分析和偏微分方程中的等变Borel提升 显示英文标题

标题： Equivariant Borel liftings in complex analysis and PDE

摘要： 我们建立了复分析和偏微分方程中几个经典定理的Borel等变类似物。 起点是一个关于整函数的等变Weierstrass定理：存在一个Borel映射，将每个非周期性正除子$d$分配给一个整函数$f_d$，其零点除子为$\mathrm{div}(f_d)=d$，并且与平移交换，$f_{d-w}(z)=f_d(z+w)$。 我们还研究了分布拉普拉斯算子、热算子和空间上的光滑函数的$\bar{\partial}$-算子的等变Borel右逆的存在性。 我们证明了这些映射在范围的自由部分上存在Borel等变逆（对于热算子，这在任何不变概率测度下去除一个零测集后成立）。 一般来说，自由性假设不能省略，Borel性也不能加强为连续性。 我们的积极结果来自于一个建立等变Borel提升存在性的充分条件的定理。 两个关键要素是Runge型逼近定理和Borel toast的存在性，这些Borel toast是遍历理论中Rokhlin塔的Borel类似物。 显示英文摘要

摘要： We establish Borel equivariant analogues of several classical theorems from complex analysis and PDE. The starting point is an equivariant Weierstrass theorem for entire functions: there exists a Borel mapping which assigns to each non-periodic positive divisor $d$ an entire function $f_d$ with divisor of zeros $\mathrm{div}(f_d)=d$ and which commutes with translation, $f_{d-w}(z)=f_d(z+w)$. We also examine the existence of equivariant Borel right inverses for the distributional Laplacian, the heat operator, and the $\bar{\partial}$-operator on the space of smooth functions. We demonstrate that Borel equivariant inverses for these maps exist on the free part of the range (for the heat operator, this holds up to the removal of a null set with respect to any invariant probability measure). In general, the freeness assumptions cannot be omitted and Borelness cannot be strengthened to continuity. Our positive results follow from a theorem establishing sufficient conditions for the existence of equivariant Borel liftings. Two key ingredients are Runge-type approximation theorems and the existence of Borel toasts, which are Borel analogues of Rokhlin towers from ergodic theory.