Affiliation:
1. Department of Mathematics, Shanghai University, Shanghai 200444 , Shanghai , China
Abstract
Abstract
In this article, we are concerned with the existence of weak
C
1
,
1
{C}^{1,1}
solution of the
k
k
-Hessian equation on a closed Hermitian manifold under the optimal assumption of the function in the right-hand side of the equation. The key points are to show the weak
C
1
,
1
{C}^{1,1}
estimates. We prove a Cherrier-type inequality to obtain the
C
0
{C}^{0}
estimate, and the complex Hessian estimate is proved by using an auxiliary function, which was motivated by Hou et al. and Tosatti and Weinkove. Our result generalizes the Kähler case proved by Dinew et al.
Subject
General Mathematics,Statistical and Nonlinear Physics
Reference30 articles.
1. Z. Blocki, Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds, Math. Z. 244 (2003), no. 1, 153–161.
2. Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1735–1756.
3. P. Cherrier, Équations de Monge-Ampère sur les variétés Hermitiennes compactes, Bull. Sci. Math. (2) 111 (1987), no. 4, 343–385.
4. J. Chu and N. McCleerey, Fully non-linear degenerate elliptic equations in complex geometry, J. Funct. Anal. 281 (2021), no. 9, Paper No. 109176, 45.
5. J. Chu, V. Tosatti, and B. Weinkove, The Monge-Ampère equation for non-integrable almost complex structures, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 7, 1949–1984.