Poincaré- and Sobolev- Type Inequalities for Complex m-Hessian Equations

Abstract

By using quasi-Banach techniques as key ingredient we prove Poincaré-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 locki's integrability conjecture for msubharmonic functions. Mathematics Subject Classification. Primary 35J60, 46E35, 26D10; Secondary 32U05, 31C45.

On a Family of Quasimetric Spaces in Generalized Potential Theory

“…Proof First assume that the sequence u j is decreasing. Then by Proposition 3.4 (6), and the monotone convergence theorem, we get…”

Trace Inequalities, isocapacitary inequalities and regularity of the complex Hessian equations

“…Theorem 4.1. [Hou,AC20] Suppose Ω is a pseudoconvex domain with smooth boundary, and 1 ≤ l < k ≤ n. Then there exists a uniform constant C > 0 depending on k, l and Ω such that…”

“…On the other hand, people also discovered that they are equivalent to isoperimetric and isocapacitary inequalities [Ma]. Despite the classical Sobolev and Moser-Trudinger inequalities, the analogous inequalities for a series of fully nonlinear equations with variational structure have been developed, including both real and complex Hessian equations [W1,TrW,TiW,AC20]. In particular, the Moser-Trudinger type inequality for the complex Monge-Ampère equations has been established [BB,C19,GKY,AC19,WWZ1].…”

“…By the Sobolev inequality for the complex Hessian equation with respective to the Lebesgue measure [AC20], we get…”