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 Lp-Calderón–Zygmund inequality: ‖Hessφ‖Lp≤C(‖φ‖Lp+‖Δφ‖Lp),∀φ∈Cc∞(M).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 De Philippis and Núñez–Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the Lp-gradient estimates and the Lp-Calderón–Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.

Marini, L., Veronelli, G. (2021). The Lp -Calderón–Zygmund inequality on non-compact manifolds of positive curvature. ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, 60(2), 253-267 [10.1007/s10455-021-09770-9].

The Lp -Calderón–Zygmund inequality on non-compact manifolds of positive curvature

Marini L.
;
Veronelli G.
2021

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 Lp-Calderón–Zygmund inequality: ‖Hessφ‖Lp≤C(‖φ‖Lp+‖Δφ‖Lp),∀φ∈Cc∞(M).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 De Philippis and Núñez–Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the Lp-gradient estimates and the Lp-Calderón–Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.
Articolo in rivista - Articolo scientifico
Calderón–Zygmund inequalities; Lp-gradient estimates; Positive sectional curvature; Sobolev Spaces on manifolds;
English
21-mag-2021
2021
60
2
253
267
none
Marini, L., Veronelli, G. (2021). The Lp -Calderón–Zygmund inequality on non-compact manifolds of positive curvature. ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, 60(2), 253-267 [10.1007/s10455-021-09770-9].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/319214
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