The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an L q,p -Sobolev inequality (2 ≤ p, q ≤ p ∗ ), provided the negative part of its Ricci tensor is small (in a suitable spectral sense). In the route, we discuss potential theoretic properties of the ends of a manifold enjoying an L q,p -Sobolev inequality.
Pigola, S., Setti, A., Troyanov, M. (2014). The connectivity at infinity of a manifold and Lq,p-Sobolev inequalities. EXPOSITIONES MATHEMATICAE, 32(4), 365-383 [10.1016/j.exmath.2013.12.006].
The connectivity at infinity of a manifold and Lq,p-Sobolev inequalities
Pigola, S
;
2014
Abstract
The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an L q,p -Sobolev inequality (2 ≤ p, q ≤ p ∗ ), provided the negative part of its Ricci tensor is small (in a suitable spectral sense). In the route, we discuss potential theoretic properties of the ends of a manifold enjoying an L q,p -Sobolev inequality.
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