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 Błocki’s integrability conjecture for m-subharmonic functions.
Highlights
In 1985, Caffarelli, Nirenberg, Spruck introduced the so called real k-Hessian operator, Sk, in bounded domains in Rn, n ≥ 2, 1 ≤ k ≤ n [17]
By these definitions we get that the 1-Hessian operator is the classical Laplace operator defined on 1-admissible functions that are just the subharmonic functions
The extension of m-subharmonic functions and the complex m-Hessian operator to non-smooth admissible functions was done by Blocki in 2005 [16]
Summary
In 1985, Caffarelli, Nirenberg, Spruck introduced the so called real k-Hessian operator, Sk, in bounded domains in Rn, n ≥ 2, 1 ≤ k ≤ n [17].
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