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

We prove some L a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of n and on compact Kähler manifolds. We also show optimal Lp integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of Błocki. Finally we obtain a local regularity result for W2,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.

Hessian equation, a priori estimate, pluripotential theory
Mathematical Subject Classification 2010
Primary: 32U15
Secondary: 32U05
Received: 6 February 2013
Accepted: 27 November 2013
Published: 7 May 2014
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
Sławomir Kołodziej
Department of Mathematics and Computer Science
Jagiellonian University
ul Lojasiewicza 6
30-348 Krakow