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

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.