Spectral properties of the canonical solution operator to dbar

The Bergman projection and weighted $C^k$ estimates for the canonical solution to \dbar on non-smooth domains

We apply integral representations for functions on non-smooth strictly pseudoconvex domains, the Henkin-Leiterer domains, to derive weighted $C^k$ estimates for the component of a given function, $f$, which is orthogonal to holomorphic functions in terms of $C^k$ norms of $\mdbar f$. The weights are powers of the gradient of the defining function of the domain.

Some spectral properties of the canonical solution operator to $\bar\partial$ on weighted Fock spaces

We characterize the Schatten class membership of the canonical solution operator to $\bar\partial$ acting on $L^2(e^{-2ϕ})$, where $ϕ$ is a subharmonic function with $Δϕ$ a doubling measure. The obtained characterization is in terms of $Δϕ$. As part of our approach, we study Hankel operators with anti-analytic symbols acting on the corresponding Fock space of entire functions in $L^2(e^{-2ϕ})$.

亚纯内函数与典范系统的唯一性定理

Uniqueness theorems for meromorphic inner functions and canonical systems

本文研究上半平面亚纯内函数的唯一性，利用谱、导数值与Clark测度等数据，推广Borg-Levinson两谱定理并推动典范Hamilton系统逆谱理论。

Degenerating Hermitian metrics, canonical bundle and spectral convergence

Let $(M,J)$ be a compact complex manifold of complex dimension $m$ and let $g_s$ be a one-parameter family of Hermitian forms on $M$ that are smooth and positive definite for each fixed $s\in (0,1]$ and that somehow degenerates to a Hermitian pseudometric $h$ for $s$ tending to $0$. In this paper under rather general assumptions on $g_s$ we prove various spectral convergence type theorems for the family of Hodge-Kodaira Laplacians $Δ_{\overline{\partial},m,0,s}$ associated to $g_s$ and acting on the canonical bundle of $M$. In particular we show that, as $s$ tends to zero, the eigenvalues, the…

Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. In this paper we are interested in the Dolbeault operator acting on the space of $L^2$ sections of the canonical bundle of $reg(X)$, the regular part of $X$. More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2Ω^{m,0}(reg(X),h)\rightarrow L^2Ω^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\overline{\partial}_{m,0}:L^2Ω^{m,0}(reg(X),h)\rightarrow L^2Ω^{m,1}(reg(X),h)$ where the domain of the latter operator is $Ω_c^{m,0}(reg(X))$. We establish various properties such as closed range of $\…

Convolution theorems associated with quaternion linear canonical transform and applications

Novel types of convolution operators for quaternion linear canonical transform (QLCT) are proposed. Type one and two are defined in the spatial and QLCT spectral domains, respectively. They are distinct in the quaternion space and are consistent once in complex or real space. Various types of convolution formulas are discussed. Consequently, the QLCT of the convolution of two quaternionic functions can be implemented by the product of their QLCTs, or the summation of the products of their QLCTs. As applications, correlation operators and theorems of the QLCT are derived. The proposed convoluti…