2025/10/06 by Pawlaschyk, Thomas
#26B25 #32F10 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.05009
In the spirit of Lelong and Bochner, we show that an upper semi-continuous function defined on a open tube set Ω=ω+ iℝn in ℂn, where ω is an open set in ℝn, 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 ℂn if and only if ω is a real q-convex set in ℝn, 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.