A priori estimates for complex Hessian equations

By S. Dinew, S. Kołodziej. 2011

160 citations recorded by the source corpus.

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 K\"ahler manifolds. We also show optimal $L^p$ integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of Blocki. 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f04018149170c49185fb499e2674a8631d1857cd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Hessian measures on m-polar sets and applications to the complex Hessian equations (2017, background): “Now as in [5,6] we define the complex Hessian operator for locally bounded msubharmonic functions as follows.”
A second order estimate for general complex Hessian equations (2015, background): “The corresponding problem on compact Kähler or Hermitian manifolds has also been studied extensively; see, for example, [Dinew and Kołodziej 2014; Hou 2009; Kołodziej and Nguyen 2016; Lu and Nguyen 2015; Zhang 2010].”
Degenerate complex Hessian equations on compact K\"ahler manifolds (2014, background): “ Fωn−(ω+ ddcq)m∧ ωn−m>ε, at x0 for some positive constant ε>0. Since F is continuous, by shrinking Ba little bit we can assume that (2.2) holds for every point in B. Fix δ>0. It follows from [15], [16] that we can ﬁnd uδ∈ SHm(X,ω) ∩ C∞(X) such that sup X |uδ−u| <δr2/2 ,sup X |Fδ−F| <δ”