Carleson Potentials and the Reproducing Kernel Thesis for Embedding Theorems

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.

Hankel算子的强再生核定理与Garsia范数

The strong reproducing kernel thesis for Hankel operators and the Garsia norm

本文针对再生核希尔伯特空间上的有界算子提出强再生核定理（SRKT）的判定问题，并具体研究Hardy空间上Hankel算子满足SRKT的条件，同时探讨BMO单位球在Garsia范数下的几何性质。

加权 Hardy 空间上 Hankel 算子的再生核命题

Reproducing Kernel Thesis of Hankel Operators on Weighted Hardy Spaces

本文研究了带 Muckenhoupt 权的两加权空间间 Hankel 算子的有界性，提出双权版 Garsia 范数，证明在再生核上测试足以判定算子有界性，并证明了 Carleson 嵌入定理的双权版本。

A Reproducing Kernel Thesis for Operators on $\ell^2$--valued Bergman-type Function Spaces

In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on $\ell^2$--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their $\ell^2$--valued versions. In particular, the results in this paper apply to the weighted $\ell^2$--valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.

Kernel estimate and capacity in Dirichlet type spaces

Let $μ$ be a positive finite measure on the unit circle. The Dirichlet type space $\mathcal{D}(μ)$, associated to $μ$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $μ$. First, we give an estimate of the norm of the reproducing kernel $k^μ$ of $\mathcal{D}(μ)$. Next, we study the notion of $μ$-capacity associated to $\mathcal{D}(μ)$, in the sense of Beurling--Deny. Namely, we give an estimate of $μ$-capacity of arcs in terms of the norm of $k^μ$. We also provide a new condition on closed sets to be $μ$-p…

Modulation invariant operators

The first three results in this thesis are motivated by a far-reaching conjecture on boundedness of singular Brascamp-Lieb forms. Firstly, we improve over the trivial estimate for their truncations, thus excluding potential trivial counterexamples. Secondly, in a dyadic model we give a partial result in the trilinear, 2-dimensional case when one of the functions depends only on one variable. Thirdly, we estimate the multidimensional version of the polynomial Carleson operator, whose boundedness would also be a consequence of the conejcture. The final pair of results concerns directional square…