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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.

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)?