Vol. 7, No. 1, 2014

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A priori estimates for complex Hessian equations

Sławomir Dinew and Sławomir Kołodziej

Vol. 7 (2014), No. 1, 227–244
Abstract

We prove some ${L}^{\infty }$ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of ${ℂ}^{n}$ and on compact Kähler manifolds. We also show optimal ${L}^{p}$ integrability for $m$-subharmonic functions with compact singularities, thus partially confirming a conjecture of Błocki. Finally we obtain a local regularity result for ${W}^{2,p}$ solutions of the real and complex Hessian equations under suitable regularity assumptions on the right-hand side. In the real case the method of this proof improves a result of Urbas.

Keywords
Hessian equation, a priori estimate, pluripotential theory
Primary: 32U15
Secondary: 32U05
Milestones
Received: 6 February 2013
Accepted: 27 November 2013
Published: 7 May 2014
Authors
 Sławomir Dinew Mathematics Department Rutgers Unversity Newark, NJ 07102 United States Department of Mathematics and Computer Science Jagiellonian University ul. Lojasiewicza 6 30-348 Krakow Poland Sławomir Kołodziej Department of Mathematics and Computer Science Jagiellonian University ul Lojasiewicza 6 30-348 Krakow Poland