论文标题
全球calderón-zygmund不平等现象
Global Calderón-Zygmund inequalities on complete Riemannian manifolds
论文作者
论文摘要
本文是对全球$ w^{2，p} $的有效性和失败的一些结果调查。我们审查了为获得表格$ \ |获得A-Priori $ l^p $ -Hessian估算的不同方法。 \ hess（u）\ | _ {l^p} \ leq c_1 \ | u \ | _ {l^p} + C_2 \ | f \ | _ {l^p} $在$ m $上的各种几何条件下，在实际有价值的函数和歧管有价值的地图的情况下。我们还提出了明确的和有些隐式的反例，表明通常，即使存在较低的部分曲率结合，这种整体不平等也可能无法保持。表格$ \ |的梯度估计的ro。| \ nabla u \ | _ {l^{p}} \ leq c_1 \ | u \ | _ {l^p} + C_2 \ |还讨论了f \ | _ {l^p} $，以及与$ l^{p} $的连接 - 也讨论了黑森估计。
This paper is a survey of some recent results on the validity and the failure of global $W^{2,p}$ regularity properties of smooth solutions of the Poisson equation $Δu = f$ on a complete Riemannian manifold $(M,g)$. We review different methods developed to obtain a-priori $L^p$-Hessian estimates of the form $\| \Hess(u) \|_{L^p} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$ under various geometric conditions on $M$ both in the case of real valued functions and for manifold valued maps. We also present explicit and somewhat implicit counterexamples showing that, in general, this integral inequality may fail to hold even in the presence of a lower sectional curvature bound. The rôle of a gradient estimate of the form $\| \nabla u \|_{L^{p}} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$, and its connections with the $L^{p}$-Hessian estimate, are also discussed.
