• Corpus ID: 246431215

Gradient estimate for complex Monge-Ampere equation with continuous right hand side

@inproceedings{Chen2022GradientEF,
  title={Gradient estimate for complex Monge-Ampere equation with continuous right hand side},
  author={Xiuxiong Chen and Jingrui Cheng},
  year={2022}
}

• Xiuxiong Chen, Jingrui Cheng
• Published 31 January 2022
• Mathematics

(1) The deep work of Kolodziej which states that if e ∈ L(M) for some p > 1, then φ ∈ C for some α ∈ (0, 1). (2) C2,α estimate when the right hand side F ∈ C. For real Monge Ampère equation, this goes back to Caffarelli[2]. For complex Monge-Ampère equation, following the theory of Evans-Krylov(see [15] for details on extension to complex setting), we know [φ]C2,α′ (M,g) is uniformly bounded, for each α ′ < α, if ∆φ is uniformly bounded. (3) The important theorem of Chen-He [5] which proves that if F ∈W 1,p for p > 2n, then we have n+∆φ < C. All the above results either estimate the C0 bound(or continuity bound of φ), or estimate the complex Hessian of φ. In this note, we will obtain estimates which, in a sense, are in between of these results. Namely we show that it is possible to obtain estimate for the gradient of φ, when the right hand side F is only assumed to be have certain continuity conditions. Let ω(r) be the modulus of continuity of F . That is, for r > 0, define ω(r) = supx,y∈M,d(x,y)<r |F (x)−F (y)|. Here d(x, y) is the distance function