Sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds

TL;DR

This paper establishes sharp $L^$ bounds for solutions to fully non-linear elliptic equations on compact complex manifolds, improving previous results and providing explicit examples of the bounds' limitations.

Contribution

It introduces new $L^$ estimates for degenerate and non-degenerate cases on Ke4hler and Hermitian manifolds, respectively, advancing the understanding of these complex equations.

Findings

Oscillation of solutions controlled by $L^1( ext{log}L)^n( ext{loglog}L)^r$ norm

Improved $L^$ estimates over previous work by Guo-Phong-Tong

Explicit example showing limits of the estimates when $r  n-1$

Abstract

We study the sharp $\mathrm{L}^\infty$ estimates for fully non-linear elliptic equations on compact complex manifolds. For the case of K\"ahler 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-Amp\`ere equations, our proof relies on an inequality of H\"older-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.