Green's function estimates for compact K\"ahler manifolds and applications

Recent works of Guo-Phong-Song-Sturm established for compact K\"ahler manifolds (even for K\"ahler spaces of specific singularities) a variety of geometric estimates depending on an upper bound of $L^{1+\epsilon}$ or $L^1(\log L)^{n+\epsilon}$ norms of the volume density but not on any curvature bound, in which a key ingredient is a uniform integral estimate for Green's function. Motivated by their results and further applications, in this paper we shall prove an improved (nearly optimal) integral estimate for Green's function under $L^{1+\epsilon}$ volume density condition, and then apply it to obtain improved global geometric estimates. For instance, one of our results states that the $k$th eigenvalue of Laplacian operator $\lambda_k\ge c\cdot k^{\frac{1}{n}}(\log k)^{-3}$, where $n$ is the complex dimension of the K\"ahler manifold and $c$ depends on $n$ and $L^{1+\epsilon}$ norm of the volume density. Also, our results can be applied to the long-time or volume-noncollapsing finite-time K\"ahler-Ricci flow on compact K\"ahler manifolds and to a general K\"ahler family to further extend previous works of Guo-Phong-Song-Sturm, Guedj-T\^o and Vu.