Duality of capacities and Sobolev extendability in the plane

One sided extendability and p-continuous analytic capacities

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, $p \in \{ 0, 1, 2, \cdots \} \cup \{ \infty \}$, for compact or closed sets in $\mathbb{C}$. We use these capacities in order to characterize the removability of singularities of functions in the spaces $A^p$.

Extendability and the $\overline \partial$ Operator on the Hartogs Triangle

In this paper it is shown that the Hartogs triangle $\mathbf T$ in $\mathbf C^2$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $\mathbf T$.

复Sobolev空间中的Alexander-Taylor不等式

Alexander-Taylor's inequality for capacities in complex Sobolev spaces

本文在复Sobolev空间中建立了Alexander-Taylor容量与功能容量之间的精确不等式，拓展了Kähler流形上的容量理论。

复流形上的高阶复 Sobolev 空间

Higher complex Sobolev spaces on complex manifolds

本文研究复流形上的高阶复 Sobolev 空间及其泛函容量，证明其 Moser-Trudinger 不等式，并探讨其与复 Monge-Ampère 方程的关系。