Weighted energy class of $m$-subharmonic functions

TL;DR

This paper introduces a new class of m-subharmonic functions with weighted energy and solves related complex Hessian equations, advancing the understanding of subextension problems with preserved weighted measures.

Contribution

It defines the class \(\mathcal{E}_{m,F}(\Omega)\), solves complex m-Hessian equations within it, and studies subextension with unchanged weighted Hessian measures, extending prior results.

Findings

Defined the class \(\mathcal{E}_{m,F}(\Omega)\) for m-subharmonic functions.

Solved complex m-Hessian equations in this class.

Analyzed subextension with preserved weighted Hessian measures.

Abstract

In this paper, we introduce the class $\mathcal{E}_{m,F}(\Omega)$ and solve complex $m$-Hessian equations in the class $\mathcal{E}_{m,F}(\Omega)$. Afterthat, we study subextension in the class $\mathcal{E}_{m,F}(\Omega)$ with the weighted Hessian measure of subextension unchanged. This is an extensive version of the result in [24] in the case when the smaller domain is relatively compact inside the large one.