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Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose that $(M,g)$ admits a bounded geometry.

Q Can we show that for $k-\frac{m}{p}\geq l-\frac{m}{q}$, we have a continuous embedding $$L^p_k \hookrightarrow L^q_l?$$

PS: I think it is true, at least for the equality as in the Aubin's book. I do not how to show the embedding for the inequality case(assuming the equality case is true)?

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    $\begingroup$ I think this is Theorem 3.14, p. 31 of Hebey, Sobolev Spaces on Riemannian Manifolds, with the hypothesis that (a) Ricci is bounded below and (b) volumes of unit balls are bounded below by a positive constant. But the precise statement is a little tricky to dig out of the book, so I am not sure. $\endgroup$
    – Ben McKay
    Commented Feb 19, 2019 at 13:20
  • $\begingroup$ In general, Sobolev inequalities on manifolds can be to much to ask. That inequality would imply that the volume of the balls grow faster that $r^n$. In particular it would be false in spaces like $\mathbb{R}^n \# \mathbb{R}^n$. Local or scale-invariant versions of the Sobolev inequalities are much more stable. I suggest you have a look at Saloff-Coste "Aspect of Sobolev type inequalities". There is also a nice survey "Sobolev inequalities in familiar and unfamiliar settings" of the same author in a book edited by Maz'ya. $\endgroup$
    – Adrián González Pérez
    Commented Feb 22, 2019 at 11:10

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