arXiv subjects

Thomas Pawlaschyk

Publications and source records attributed to Thomas Pawlaschyk.

8 recordsLinked to original sources

arXiv2026

The Perron-Bremermann envelope for $q$-plurisubharmonic functions on unbounded domains in $\mathbb{C}^n$

Let $D$ be an unbounded domain in $\mathbb{C}^n$, and let $f$ be a bounded continuous function prescribed on the boundary of $D$. We show that, if $D$ has $r$-peak points on its boundary and is of bounded type, $f$ extends to a maximal bounded continuous function $F$ on $\overline{D}$ that is $q$-plurisubharmonic and $(n-q-1)$-plurisuperharmonic (i.e., $(dd^c F)^n=0$) on $D$, and that coincides with the Perron-Bremermann envelope created with respect to bounded $q$-plurisubharmonic functions on $\overline{D}$.

math.CV↗

arXiv2025

On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$

In the spirit of Lelong and Bochner, we show that an upper semi-continuous function defined on a open tube set $Ω=ω+ i\mathbb{R}^n$ in $\mathbb{C}^n$, where $ω$ is an open set in $\mathbb{R}^n$, and which is invariant in its imaginary part, is $q$-plurisubharmonic on $Ω$ (in the sense of Hunt and Murray) if and only if it is real $q$-convex on $ω$, i.e., it admits the local maximum property with respect to affine linear functions on real $(q+1)$-dimensional affine subspaces. From this, we conclude that, for $a>0$, the set $ω+i(-a,a)^n$ is $q$-pseudoconvex in $\mathbb{C}^n$ if and only if $ω$ is a real $q$-convex set in $\mathbb{R}^n$, i.e., $ω$ admits a real $q$-convex exhaustion function on $ω$. We apply these results to complements of graphs of affine linear maps and to Reinhardt domains.

math.CV↗

arXiv2025

A Selection Theorem for the Carathéodory Kernel Convergence of Pointed Domains

We present a selection theorem for domains in $\mathbb{C}^n$, $n\ge 1$, which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carathéodory. Not only is this analogous to the well-known Blaschke selection theorem for compact convex sets, but it fits better in the study of normal families of holomorphic maps with varying domains and ranges.

math.CV↗

arXiv2025

Foliations of continuous q-pseudoconcave graphs

We show that for $k = 0, 1$ the graph of a continuous mapping $f:D \to \mathbb{R}^k\times\mathbb{C}^p$, defined on a domain $D$ in $\mathbb{C}^n\times\mathbb{R}^k$, is locally foliated by complex $n$-dimensional submanifolds if and only if its complement is $n$-pseudoconvex (in the sense of Rothstein) relatively to $(D\times\mathbb{R}^k)\times\mathbb{C}^p\subset \mathbb{C}^{n}\times\mathbb{C}^k\times\mathbb{C}^p$.

math.CV↗

arXiv2025

On compact sets possessing $q$-convex functions

We show that there exists a $q$-convex function in a neighborhood of a compact set $K$ in a complex manifold $\mathcal{M}$ if and only if the $q$-nucleus of this compact set is empty. The latter can be characterized as the maximal $q$-pseudoconcave subset of $K$, i.e., a subset of $K$ containing all other compact $q$-pseudoconcave subsets in $K$.

math.CV↗

arXiv2020

The Bergman-Shilov boundary for subfamilies of $q$-plurisubharmonic functions

We introduce a notion of the Bergman-Shilov (or Shilov) boundary for some subclasses of upper-semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of that subclass attain their maximum. For certain subclasses with simple structure one can show the existence and uniqueness of the Shilov boundary. Then we provide its relation to the set of peak points and establish Bishop-type theorems. As an application we obtain a generalization of Bychkov's theorem which gives a geometric characterization of the Shilov boundary for $q$-plurisubharmonic functions on convex bounded domains. In the case of bounded pesudoconvex domains with smooth boundary we also show that some parts of the Shilov boundary for $q$-plurisubharmonic functions are foliated by $q$-dimensional complex submanifolds.

math.CV↗

arXiv2019

Engaging students in conjecturing through homework in Real Analysis and Differential Equations

In this note we report on an implementation of discovery-oriented problems in courses on Real Analysis and Differential Equations. We explain a type of task-design that gives students the opportunity to conjecture, refute and prove. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework assignments. In addition to several concrete examples we also discuss feedback and assessment outcomes of our students.

math.HO↗

arXiv2018

On convex hulls and pseudoconvex domains generated by $q$-plurisubharmonic functions, part III

We characterise in this work the $q$-plurisubharmonic functions in terms of the theory of viscosity solutions. We show that an upper semicontinuous function is $q$-plurisubharmonic if and only if its complex Hessian has at most $q$ strictly negative eigenvalues in the viscosity sense. This characterisation is then used to prove that the supremum convolution of a (strictly) $q$-plurisubharmonic function is again (strictly) $q$-plurisubharmonic on a maybe different set of definition. Finally, we use the supremum convolution to deduce a new characterisation for the $q$-pseudoconvex subsets in $\mathbb{C}^n$.

math.CV↗