论文标题
$ \ mathbb {c}^n $在单位球上的全态函数的边界痕迹
Boundary Traces of Holomorphic Functions on the Unit Ball in $\mathbb{C}^n$
论文作者
论文摘要
这是一个经典定理，如果一个函数沿单位圆的边界可以集成，那么当且仅当其非负整数的傅立叶系数为零时，该函数是敞开盘上全态函数的非区别限制。在本文中，我们通过证明在单位球体上的可集成函数来将此结果推广到较高的复合维度，仅当满足两个特定的积分方程族时，它是空心单位球上holomorthic函数的``边界跟踪''。为此，我们使用耐力空间的理论以及不变的泊松和库奇积分。本文以一种旨在欢迎那些参加复杂分析课程但不一定是该领域专家的人的风格写作。
It is a classical theorem that if a function is integrable along the boundary of the unit circle, then the function is the nontangential limit of a holomorphic function on the open disc if and only if its Fourier coefficients for nonnegative integers are zero. In this article we generalize this result to higher complex dimensions by proving that for an integrable function on the unit sphere, it is ``boundary trace'' of a holomorphic function on the open unit ball if and only if two particular families of integral equations are satisfied. To do this, we use the theory of Hardy spaces as well as the invariant Poisson and Cauchy integrals. This article is written in a style that is meant to be welcoming to those who have taken a course in complex analysis but who are not necessarily experts in the field.
