Sharp L^∞ estimates for fully non-linear elliptic equations on compact complex manifolds

2024/09/08 by Qiao, Yuxiang
#32Q99 #35J60 (Primary) 32Q15 #35J70 (Secondary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.05157

Abstract

We study the sharp L^∞ estimates for fully non-linear elliptic equations on compact complex manifolds. For the case of Kähler manifolds, we prove that the oscillation of any admissible solution to a degenerate fully non-linear elliptic equation satisfying several structural conditions can be controlled by the L¹(\logL)ⁿ(log\logL)^r(r>n) norm of the right-hand function (in a regularized form). This result improves that of Guo-Phong-Tong. In addition to their method of comparison with auxiliary complex Monge-Ampère equations, our proof relies on an inequality of Hölder-Young type and an iteration lemma of De Giorgi type. For the case of Hermitian manifolds with non-degenerate background metrics, we prove a similar L^∞ estimate which improves that of Guo-Phong. An explicit example is constucted to show that the L^∞ estimates given here may fail when r\leqslant n-1. The construction relies on a gluing lemma of smooth, radial, strictly plurisubharmonic functions.

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