Poincar\'e- and Sobolev- type inequalities for complex $m$-Hessian equations

TL;DR

This paper establishes Poincaré and Sobolev inequalities for m-subharmonic functions with finite energy, providing partial support for B extlocki's integrability conjecture in complex analysis.

Contribution

It introduces new inequalities for m-subharmonic functions using quasi-Banach techniques, advancing understanding of their integrability properties.

Findings

Proves Sobolev-type inequalities for m-subharmonic functions

Provides partial confirmation of B extlocki's integrability conjecture

Uses quasi-Banach techniques as a key methodological tool

Abstract

By using quasi-Banach techniques as key ingredient we prove Poincar\'e- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of B\l ocki's integrability conjecture for $m$-subharmonic functions.