Refined Sobolev inequalities on manifolds with ends

Abstract

By considering a suitable Besov type norm, we obtain refined Sobolev inequalities on a family of Riemannian manifolds with (possibly exponentially large) ends. The interest is twofold: on one hand, these inequalities are stable by multiplication by rapidly oscillating functions, much as the original ones [5], and on the other hand our Besov space is stable by spectral localization associated to the Laplace-Beltrami operator (while L spaces, with p 6= 2, are in general not preserved by such localizations on manifolds with exponentially large ends). We also prove an abstract version of refined Sobolev inequalities for any selfadjoint operator on a measure space (Proposition 1). For functions u on Rn, n ≥ 1, and p ≥ 2 real, the homogeneous and inhomogeneous Sobolev estimates ||u||Lp(Rn) ≤ C||(−∆)u||L2(Rn) =: ||u||Ḣsp ≤ C||(1−∆)u||L2(Rn) =: ||u||Hsp , sp = n 2 − n p , (1) are a very well known tool to control ||u||Lp when u is sufficiently smooth. The homogeneous version is sharp with respect to all scalings and the inhomogeneous one is sharp with respect to high frequency scalings (i.e. u(x) 7→ u(λx) with λ ≥ 1). The drawback of these estimates is to behave badly under the multiplication by characters. The usual counterexample (see e.g. [1, subsection 1.3.2]) is to consider ue(x) = e i e x·ηφ(x), (2) with φ 6= 0 in the Schwartz space S(Rn), η 6= 0 in Rn, and to observe that ||ue||Lp = ||φ||Lp , ||ue||Ḣsp ∼ ||ue||Hsp ∼ e −sp |η|p ||φ||L2 , for which the Sobolev estimates alone provide only the very bad estimate ||ue||Lp . e−sp . Notice that considering the homogeneous or inhomogenous Sobolev norm is irrelevant here. ∗Jean-Marc.Bouclet@math.univ-toulouse.fr †sire@cmi.univ-mrs.fr