Cross theorem

Lefschetz theorem for holomorphic one-forms on weakly 1-complete manifolds

For a holomorphic one-form $\mathbfξ$ on a weakly 1-complete manifold $X$ with certain properties, we discussed the connectivity of the pair $(\hat{X}, F^{-1}(z))$, where $π: \hat{X} \to X$ is a covering map and $dF=π^*\mathbfξ$. We also discussed the criteria about when such a manifold $X$ admits a proper holomorphic mapping onto a Riemann surface.

Boundary cross theorem in dimension 1

Let $X, Y$ be two complex manifolds of dimension 1 which are countable at infinity, let $D\subset X,$ $ G\subset Y$ be two open sets, let $A$ (resp. $B$) be a subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in addition that $D$ (resp. $G$) is {\it Jordan-curve-like on $A$} (resp. $B$) and that $A$ and $B$ are {\it of positive length}. We determine the "envelope of holomorphy" $\hat{W}$ of $W$ in the sense that any function locally bounded on $W,$ measurable on $A\times B,$ and separately holomorphic on $(A\times…

Boundary cross theorem in dimension 1 with singularities

Let $D$ and $G$ be copies of the open unit disc in $\C,$ let $A$ (resp. $B$) be a measurable subset of $\partial D$ (resp. $\partial G$), let $W$ be the 2-fold cross $\big((D\cup A)\times B\big)\cup \big(A\times(B\cup G)\big),$ and let $M$ be a relatively closed subset of $W.$ Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\cap (A\times B)=\varnothing.$ We determine the "envelope of holomorphy" $\hat{W\setminus M}$ of $W\setminus M$ in the sense that any function locally bounded on $W\s…

Factorisation of conformal maps on finitely connected domains

Let $U$ be a multiply connected domain of the Riemann sphere $\hat{C}$ whose complement $\hat{C}\setminus U$ has $N<\infty$ components. We show that every conformal map on $U$ can be written as a composition of $N$ maps conformal on simply connected domains. This improves on a recent result of D.E. Marshall, but our proof uses different ideas, and involves the uniformisation theorem for Riemann surfaces.

A Paley-Wiener Type Theorem for Singular Measures on $\mathbb{T}$

For a fixed singular Borel probability measure $μ$ on $\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(μ)$. The first characterization is given in terms of criteria for sampling functions of the form $\hat{f}$ when $f \in L^2(μ)$. The second characterization is given in terms of criteria for interpolation of bounded sequences on $\mathbb{N}_{0}$ by $\hat{f}$. Both characterizations use the construction of Fourier series for $f \in L^2(μ)$ demonstrated in Herr and Weber via the Kaczmarz algorithm and classical results concern…

量子十字的谱常数与环域的渐近锐界

The spectral constant for the quantum cross and asymptotically sharp bounds for annuli

本文研究量子环域上解析函数求值的谱常数，通过将问题退化为量子十字情形，并利用有理扩张技巧证明当r趋于无穷时谱常数极限为2。

Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3

Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hatθ$ so that $\hatθ$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} \hat{X}, \hatθ)$ is a pseudohermitian $3$-manifold. Let $G_p \in C^{\infty} (\hat{X} \setminus \{p\})$, $G_p > 0$, with $G_p(x) \sim \vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat{X}$. Consider the new pseudohermitian $…

Central Limit theorem for spectral Partial Bergman kernels

Partial Bergman kernels $Π_{k, E}$ are kernels of orthogonal projections onto subspaces $\mathcal{k} \subset H^0(M, L^k)$ of holomorphic sections of the $k$th power of an ample line bundle over a Kahler manifold $(M, ω)$. The subspaces of this article are spectral subspaces $\{\hat{H}_k \leq E\}$ of the Toeplitz quantization $\hat{H}_k$ of a smooth Hamiltonian $H: M \to \mathbb{R}$. It is shown that the relative partial density of states $\frac{Π_{k, E}(z)}{Π_k(z)} \to {1}_{\mathcal{A}}$ where $\mathcal{A} = \{H < E\}$. Moreover it is shown that this partial density of states exhibits `Erf'-as…