Asymptotic of Bergman Kernel (Master Thesis)

A reproducing kernel thesis for operators on Bergman-type function spaces

In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit polydisc and more generally to weighted Fock spaces.

The reproducing kernel thesis for lower bounds of weighted composition operators

It is shown that the property of being bounded below (having closed range) of weighted composition operators on Hardy and Bergman spaces can be tested by their action on a set of simple test functions, including reproducing kernels. The methods used in the analysis are based on the theory of reverse Carleson embeddings.

Matrix Schrödinger Operators and Weighted Bergman Kernels

The aim of the present thesis is twofold: to study the problem of discreteness of the spectrum of Schrödinger operators with matrix-valued potentials in ${\mathbb R}^d$ (Chapter 1), and to prove new pointwise bounds for weighted Bergman kernels in ${\mathbb C}^n$ (chapters 2 and 3). These two themes are connected by the observation that the weighted Bergman kernel may be effectively studied by means of the analysis of the weighted Kohn Laplacian, which in turn is unitarily equivalent to a Schödinger operator with matrix-valued potential.

复流形与轨形上Toeplitz量子化的半经典渐近展开

Semi-Classical Asymptotic Expansions for Toeplitz Quantizations on Complex Manifolds and Orbifolds

该研究引入带局部谱隙的复流形，用标度法得到埃尔米特全纯轨形线丛Bergman核的渐近展开，结合标度Bergman核与稳相公式建立Bergman核和Toeplitz算子的完整渐近展开，还构建了带伪微分算子的Toeplitz算子形变量子化。