Piecewise harmonic subharmonic functions and positive Cauchy transforms

Subharmonic Configurations and Algebraic Cauchy Transforms of Probability Measures

We study subharmonic functions whose Laplacian is supported on a null set and in connected components of of the complement to the support admit harmonic extensions to larger sets. We prove that if such a function has a piecewise holomorphic derivative, then it is locally piecewise harmonic. In generic cases it coincides locally with the maximum of finitely many harmonic functions. Moreover, we describe the support when the holomorphic derivative satisfies a global algebraic equation. The proofs follow classical patterns and our methods may also be of independent interest.

On parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in $\mathbb{C}$ with respect to the perturbation parameter $ε$. This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so…

有理角点处的有限叶柯西算子

Finite-sheeted Cauchy operator at rational corners

研究平面楔形区域上开口角为π的有理倍数的柯西奇异积分算子，通过覆盖变换将其分解为区间柯西变换的有限叶组合，并分析其函数空间性质和端点展开。