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- W4287604629 abstract "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 $Delta u = f$ 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 r^ole 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." @default.
- W4287604629 created "2022-07-25" @default.
- W4287604629 creator A5091469720 @default.
- W4287604629 date "2020-11-06" @default.
- W4287604629 modified "2023-10-17" @default.
- W4287604629 title "Global Calder'on-Zygmund inequalities on complete Riemannian manifolds" @default.
- W4287604629 doi "https://doi.org/10.48550/arxiv.2011.03220" @default.
- W4287604629 hasPublicationYear "2020" @default.
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