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

TL;DR

This paper characterizes $q$-plurisubharmonic functions via viscosity solutions, demonstrating that supremum convolutions preserve $q$-plurisubharmonicity and providing a new characterization of $q$-pseudoconvex subsets in complex space.

Contribution

It introduces a viscosity-based characterization of $q$-plurisubharmonic functions and applies it to show the preservation of this property under supremum convolution, also characterizing $q$-pseudoconvex sets.

Findings

Viscosity characterization of $q$-plurisubharmonic functions.

Supremum convolution preserves $q$-plurisubharmonicity.

New characterization of $q$-pseudoconvex subsets.

Abstract

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 $\mathbb{C}^n$.