We introduce the concept of Calderón–Zygmund inequalities on Riemannian manifolds. For 1<∞, these are inequalities of the form ‖Hess(u)‖Lp≤C1‖u‖Lp+C2‖Δu‖Lp, valid a priori for all smooth functions u with compact support, and constants C1≥0, C2>0. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón–Zygmund inequality (like new denseness results for second order Lp-Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.

Güneysu, B., Pigola, S. (2015). The Calderón-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds. ADVANCES IN MATHEMATICS, 281, 353-393 [10.1016/j.aim.2015.03.027].

The Calderón-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds

PIGOLA, STEFANO
2015

Abstract

We introduce the concept of Calderón–Zygmund inequalities on Riemannian manifolds. For 1<∞, these are inequalities of the form ‖Hess(u)‖Lp≤C1‖u‖Lp+C2‖Δu‖Lp, valid a priori for all smooth functions u with compact support, and constants C1≥0, C2>0. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón–Zygmund inequality (like new denseness results for second order Lp-Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.
Articolo in rivista - Articolo scientifico
Global Calderón–Zygmund inequality; Sobolev spaces; Noncompact Riemannian manifold
English
2015
281
353
393
reserved
Güneysu, B., Pigola, S. (2015). The Calderón-Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds. ADVANCES IN MATHEMATICS, 281, 353-393 [10.1016/j.aim.2015.03.027].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/271352
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