Weak solutions to the complex Hessian equation

Liouville and Calabi-Yau type theorems for complex Hessian equations

“…It was observed by B locki ( [Bl1]) that, following the ideas of Bedford and Taylor ([BT1], [BT2]), one can relax the smoothness requirement on u and develop a non linear version of potential theory for Hessian operators.…”

“…Due to their non linear structure, and similarly to the Monge-Ampère equation, the Hessian equation is considered only over a suitable subclass of functions for which ellipticity is guaranteed. These subclasses, in a sense, interpolate between subharmonic and plurisubharmonic functions and thus they are also interesting to study from potential theoretic point of view ([Bl1], [DK], [Chi]). In the Kähler manifold case the corresponding classes of functions, namely ω − m subharmonic ones, do not yield, in general, classical positive definite metrics, which distinguishes them from ω-plurisubharmonic functions.…”

“…Likewise, the theory of the complex Hessian equation is being developed in domains of C n and on compact Kähler manifolds. The case of domains in C n is to a large extent understood ( [Li], [Bl1], [DK]). In particular the classical Dirichlet problem is solvable in the smooth category ( [Li], [Bl1]) under standard regularity assumptions on the data and some convexity assumptions on the boundary of the domain.…”

A second order estimate for complex Hessian equations on a compact Kähler manifold

“…As we know, the Dirichlet problem of complex Hessian equation in C n was studied by S. Y. Li [10] and B locki [1]. The first named author [8] proved the uniqueness of the solution of (1.2); he [8] also proved the existence of a smooth admissible solution of (1.2), by assuming the nonnegativity of the orthogonal bisectional curvature of ω.…”

On $$\varvec{m}$$ m -Subharmonic Ordering of Measures

“…If Ω is a bounded domain in C n and {u j } is a sequence of locally bounded m-subharmonic functions on Ω which is decreasing to a function u ∈ SH m (Ω) ∩ L ∞ loc (Ω), then H m (u j ) converges to H m (u) in the weak*-topology (see [4]). The same conclusion holds if…”