On peak-interpolation manifolds for A(Ω) for convex domains in C^n

Duality, convexity and peak interpolation in the Drury-Arveson space

We consider the closed algebra $\mathcal{A}_d$ generated by the polynomial multipliers on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop-Carleson-Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light o…

Geometric conditions for interpolation in weighted spaces of entire functions

We use $L^2$ estimates for the $\bar\partial$ equation to find geometric conditions on discrete interpolating varieties for weighted spaces $A_p(\C)$ of entire functions such that $| f(z)|\le Ae^{Bp(z)}$ for some $A,B>0$. In particular, we give a characterization when $p(z)=e^{| z|}$ and more generally when $\ln p(e^r)$ is convex and $\ln p(r)$ is concave.

调和插值与随机行列式的 Brunn-Minkowski 定理

Harmonic Interpolation and a Brunn-Minkowski Theorem for Random Determinants

本文提出凸体的调和插值方法，证明 Brunn-Minkowski 不等式的强形式并刻画等号情形，改进 Berndtsson 的伪凸域切片体积定理，并证明随机行列式期望绝对值的次调和性。

R-范数的复插值、对偶性与叶状结构

Complex interpolation of $\mathbb{R}$-norms, duality and foliations

本文将Calderón等人的复插值方法与Alexander-Wermer-Slodkowski定理推广到实Banach空间和凸函数的插值，利用Legendre变换揭示对偶性，并给出齐次复Monge-Ampère方程解的新性质。

Extending holomorphic maps from Stein manifolds into affine toric varieties

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstnerič, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular tar…

Hyperbolicity in unbounded convex domains

We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

动量问题与谱函数

Moment Problems and Spectral Functions

本文研究了Nevanlinna-Pick插值和动量问题，证明了因果数据空间的凸性，并回顾了相关结果。

On nonlinear Rudin-Carleson type theorem

In this paper we study nonlinear interpolation problems for interpolation and peak-interpolation sets of function algebras. The subject goes back to the classical Rudin-Carleson interpolation theorem. In particular, we prove the following nonlinear version of this theorem: Let $\bar{\mathbb D}\subset \mathbb C$ be the closed unit disk, $\mathbb T\subset\bar{\mathbb D}$ the unit circle, $S\subset\mathbb T$ a closed subset of Lebesgue measure zero and $M$ a connected complex manifold. Then for every continuous $M$-valued map $f$ on $S$ there exists a continuous $M$-valued map $g$ on $\bar{\mathb…