This paper is a survey of some recent results on the validity and the failure of global regularity properties of smooth solutions of the Poisson equation on a complete Riemannian manifold . We review different methods developed to obtain a-priori -Hessian estimates of the form under various geometric conditions on 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 rôle of a gradient estimate of the form , and its connections with the -Hessian estimate, are also discussed.
@article{TSG_2019-2021__36__127_0, author = {Stefano Pigola}, title = {Global {Calder\'on{\textendash}Zygmund} inequalities on complete {Riemannian} manifolds}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {127--189}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {36}, year = {2019-2021}, doi = {10.5802/tsg.375}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.375/} }
TY - JOUR AU - Stefano Pigola TI - Global Calderón–Zygmund inequalities on complete Riemannian manifolds JO - Séminaire de théorie spectrale et géométrie PY - 2019-2021 SP - 127 EP - 189 VL - 36 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.375/ DO - 10.5802/tsg.375 LA - en ID - TSG_2019-2021__36__127_0 ER -
%0 Journal Article %A Stefano Pigola %T Global Calderón–Zygmund inequalities on complete Riemannian manifolds %J Séminaire de théorie spectrale et géométrie %D 2019-2021 %P 127-189 %V 36 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.375/ %R 10.5802/tsg.375 %G en %F TSG_2019-2021__36__127_0
Stefano Pigola. Global Calderón–Zygmund inequalities on complete Riemannian manifolds. Séminaire de théorie spectrale et géométrie, Tome 36 (2019-2021), pp. 127-189. doi : 10.5802/tsg.375. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.375/
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