arXiv subjects

J. Lebl

Publications and source records attributed to J. Lebl.

2 recordsLinked to original sources

arXiv2019

The closed range property for the $\overline{\partial}$-operator on planar domains

Let $Ω\subset\mathbb{C}$ be an open set. We show that $\overline{\partial}$ has closed range in $L^{2}(Ω)$ if and only if the Poincaré-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic conditions for the $\overline{\partial}$-operator to have closed range in $L^{2}(Ω)$. We also give a new necessary and sufficient potential-theoretic condition for the Bergman space of $Ω$ to be infinite dimensional.

math.CV↗

arXiv2018

Convexity of level lines of Martin functions and applications

Let $Ω$ be an unbounded domain in $\mathbb{R}\times\mathbb{R}^{d}.$ A positive harmonic function $u$ on $Ω$ that vanishes on the boundary of $Ω$ is called a Martin function. In this note, we show that, when $Ω$ is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition $Ω$ is symmetric, then the maximum of any Martin function along a slice $Ω\cap (\{t\}\times\mathbb{R}^d)$ is attained at $(t,0).$

math.AP↗