We construct, for \(p>n\), a concrete example of a complete non-compact n-dimensional Riemannian manifold of positive sectional curvature which does not support any \(L^p\)-Calderón–Zygmund inequality:

The proof proceeds by local deformations of an initial metric which (locally) Gromov–Hausdorff converge to an Alexandrov space. In particular, we develop on some recent interesting ideas by De Philippis and Núñez–Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the \(L^p\)-gradient estimates and the \(L^p\)-Calderón–Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.

对于\（p> n \），我们构造了一个完整的非正n维正截面曲率黎曼流形的具体示例，它不支持任何\（L ^ p \）- Calderón–Zygmund不等式：

证明是通过初始度量的局部变形进行的，该初始度量（局部地）是Gromov–Hausdorff收敛到Alexandrov空间。特别是，我们借鉴了De Philippis和Núñez–Zimbron关于紧凑流形的最新观点。作为直接的结果，我们得出\（L ^ p \）-梯度估计和\（L ^ p \）- Calderón–Zygmund不等式通常不相等，因此回答了文献中的一个未解决的问题。最后，我们的示例也为流形上Sobolev空间的不同定义的（非）等价性研究做出了贡献。