On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
Thomas Pawlaschyk
TL;DR
The paper develops a real convexity framework to generalize Lelong–Bochner-type results to q-plurisubharmonic and q-pseudoconvex contexts for tube domains. It defines real q-convex functions on open sets in real space and shows they correspond to q-plurisubharmonic functions on tube domains, yielding two main theorems: (i) a USC, imaginary-part-invariant function on a tube is q-plurisubharmonic iff its real part is real q-convex; (ii) a real q-convex ω is equivalent to ω+i(-a,a)^n being q-pseudoconvex for any a>0. The work develops robust approximation methods, Dirichlet envelopes for q-convexity, and applications to complements of affine graphs and Reinhardt domains, establishing a broad bridge between real convexity and complex pseudoconvexity with potential implications for generalized convexity in several complex variables.
Abstract
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.