On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
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| Format: | Preprint |
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2025
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| _version_ | 1866916997750587392 |
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| author | Pawlaschyk, Thomas |
| author_facet | Pawlaschyk, Thomas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$ Pawlaschyk, Thomas Complex Variables 32F10, 26B25 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. |
| title | On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$ |
| topic | Complex Variables 32F10, 26B25 |
| url | https://arxiv.org/abs/2510.05009 |