标题： 完全非线性椭圆方程在紧致复流形上的$\mathrm{L}^\infty$估计 显示英文标题

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

摘要： 我们研究紧致复流形上完全非线性椭圆方程的精确$\mathrm{L}^\infty$估计。 对于凯勒流形的情况，我们证明了满足若干结构条件的退化完全非线性椭圆方程的任何可接受解的振荡可以由右边函数的$\mathrm{L}^1(\log\mathrm{L})^n(\log\log\mathrm{L})^r(r>n)$范数（以正则化形式）控制。 这一结果改进了 Guo-Phong-Tong 的结果。 除了他们使用辅助复 Monge-Ampère 方程的比较方法外，我们的证明还依赖于一个 Hölder-Young 型不等式和一个 De Giorgi 型迭代引理。 对于具有非退化背景度量的埃尔米特流形的情况，我们证明了一个类似的$\mathrm{L}^\infty$估计，该估计改进了 Guo-Phong 的结果。 构造了一个显式例子来说明这里的$\mathrm{L}^\infty$估计在$r\leqslant n-1$时可能失效。 该构造依赖于光滑、径向、严格全纯次调和函数的粘合引理。 显示英文摘要

摘要： 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.