Global Calderón–Zygmund inequalities on complete Riemannian manifolds

Stefano Pigola

doi:10.5802/tsg.375

Stefano Pigola ¹

¹ Università degli Studi di Milano-Bicocca Dipartimento di Matematica e Applicazioni Via Cozzi 55, 20126 Milano - ITALY

Séminaire de théorie spectrale et géométrie, Volume 36 (2019-2021), pp. 127-189

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 $Δ u = f$ 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 rôle 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.

Published online: 2024-08-05

DOI: 10.5802/tsg.375

Author's affiliations:

Stefano Pigola ¹

¹ Università degli Studi di Milano-Bicocca Dipartimento di Matematica e Applicazioni Via Cozzi 55, 20126 Milano - ITALY

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     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},
     year = {2019-2021},
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     volume = {36},
     doi = {10.5802/tsg.375},
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Stefano Pigola. Global Calderón–Zygmund inequalities on complete Riemannian manifolds. Séminaire de théorie spectrale et géométrie, Volume 36 (2019-2021), pp. 127-189. doi: 10.5802/tsg.375

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