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

We study the sharp ${\mathrm{L}}^{\mathrm{\infty }}$ 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 ${\mathrm{L}}^{\mathrm{\infty }}$ estimate which improves that of Guo-Phong. An explicit example is constucted to show that the ${\mathrm{L}}^{\mathrm{\infty }}$ estimates given here may fail when $r⩽n-1$. The construction relies on a gluing lemma of smooth, radial, strictly plurisubharmonic functions.