Let $\Omega \subset \mathbb{R}^n$ be a convex polytope ($n \leq 3$). The Ritz projection is the best approximation, in the $W^{1,2}_0$-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, that the gradient at any point in $\Omega$ is controlled by the Hardy--Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of interest in the analysis of PDEs immediately follows. Among those are weighted spaces, Orlicz spaces and Lorentz spaces.

翻译：设 $\Omega \subset \mathbb{R}^n$ 为凸多面体（$n \leq 3$）。Ritz投影是在 $W^{1,2}_0$ 范数下对给定函数在有限元空间中的最佳逼近。当基于拟一致三角剖分构造此类有限元空间时，我们证明了Ritz投影的一个逐点估计，即 $\Omega$ 中任意点的梯度受该点原始函数梯度的哈代-利特尔伍德极大函数控制。由此估计可立即推导出Ritz投影在偏微分方程分析中涉及的一系列重要空间（包括加权空间、奥尔利奇空间和洛伦兹空间）上的稳定性。