论文标题
$ l^p $-calderón-Zygmund不平等现象，在非紧密曲线上
The $L^p$-Calderón-Zygmund inequality on non-compact manifolds of positive curvature
论文作者
论文摘要
我们以$ p> n $的形式构建了一个完整的非紧凑型$ n $二维的riemannian分段曲率流形的具体示例，该分段曲率不支持任何$ l^p $-calderón-zygmund不平等：\ [\ [\ [ \ forall \，φ\ in c^{\ infty} _c（m），\ qquad \ | \ operatoRatOrname {hess}φ\ | _ {l^p} \ le c（\ | c（\ |φ\ | _ ________________ {l^p}+\ | g}+\ | ex | eucum | euct | __________}） \]证明是通过（本地）Gromov-Hausdorff收敛到Alexandrov空间的初始度量的局部变形进行的。特别是，我们根据G. de Philippis和J.Núñez-Zimbron的一些最新有趣的想法进行了发展，以处理紧凑的歧管。这是一个简单的结果，我们得到$ l^p $差异估计和$ l^p $-calderón-Zygmund的不平等通常不是等效的，因此在文献中回答了一个公开的问题。最后，我们的示例还为研究sobolev空间不同定义的（非）等效性的研究做出了贡献。
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: \[ \forall\,φ\in C^{\infty}_c(M),\qquad\|\operatorname{Hess} φ\|_{L^p}\le C(\|φ\|_{L^p}+\|Δφ\|_{L^p}). \] 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 G. De Philippis and J. 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 literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.
