Sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds
Year: 2024
Type: preprint
Abstract: We study the sharp $\mathrm{L}^\infty$ 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 $\mathrm{L}^1(\log\mathrm{L})^n(\log\log\mathrm{L})^r(r>n)$ 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 $\mathrm{L}^\infty$ estimate which improves that of Guo-Phong. An explicit example is constucted to show that the $\mathrm{L}^\infty$ estimates given here may fail when $r\leqslant n-1$. The construction relies on a gluing lemma of smooth, radial, strictly plurisubharmonic functions.
Source: arXiv (Cornell University)
Author: Qiao, Yuxiang
Institution: Peking University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403617641