AI 中文总结
研究紧致Hermitian流形上复Hessian方程，通过开发多重势方法，结合弱比较原理、容量理论及非线性迭代方案，在\(f \in L^p\)且\(f \ge 0\)（\(p>1\)）时给出一致\(L^\infty\)估计，并得到相关弱解结果，扩展了该理论。
AI 中文摘要
我们针对紧致Hermitian流形上的复Hessian方程开发了一种多重势方法。在此情形下，背景度量缺乏闭性会引入挠率项，阻碍Kähler理论的直接推广。我们的主要结果是在\(f \in L^p\)且\(f \ge 0\)（其中\(p>1\)）的假设下，对方程\((\omega + dd^c u)^m \wedge \omega^{n - m} = cf\,\omega^n\)的有界\(\omega\) - \(m\) - 次调和解给出一致\(L^\infty\)估计。证明结合了带有挠率误差的弱比较原理、适用于Hermitian情形的容量理论以及控制子水平集衰减的非线性迭代方案。作为应用，我们得到了具有\(L^p\)密度的弱解的存在性、稳定性和紧致性结果。这些结果扩展了复Hessian方程多重势理论在Kähler框架之外多个方面的内容。
英文摘要
We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the Kähler theory. Our main result is a uniform $L^\infty$ estimate for bounded $ω$-$m$-subharmonic solutions of the equation \[ (ω+ dd^c u)^m \wedge ω^{n-m} = cf\,ω^n, \] under the assumption that $f \in L^p$, $f \ge 0$ for some $p>1$. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with $L^p$ densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the Kähler framework.
