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- W3165721666 abstract "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 $L^p$-Calder'on-Zygmund inequality: [ forall,varphiin C^{infty}_c(M),qquad|operatorname{Hess} varphi |_{L^p}le C(|varphi|_{L^p}+|Deltavarphi|_{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. N'u~nez-Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the $L^p$-gradient estimates and the $L^p$-Calder'on-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." @default.
- W3165721666 created "2021-06-07" @default.
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- W3165721666 date "2021-05-21" @default.
- W3165721666 modified "2023-10-16" @default.
- W3165721666 title "The $$L^p$$-Calderón–Zygmund inequality on non-compact manifolds of positive curvature" @default.
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- W3165721666 doi "https://doi.org/10.1007/s10455-021-09770-9" @default.
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