# Sobolev Multiplication on non-compact manifold

We know that for a compact Riemannian $$n$$-dim manifold $$(M,g)$$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $$L^p_k\times L^q_l⟶L^r_m$$, where $$1/r−m/n>1/p−k/m+1/q−l/n$$ and $$m\leq\min(l,k)$$.

Q If $$M$$ is non-compact, does the Sobolev Multiplication Theorem still holds?

PS:

• If $$k=l=m=0$$, we know that it is true. Could anyone give a reference for the case $$k,l,m\geq1$$?

• The problem is that can we have an embedding $$L^p_k\hookrightarrow L^{p*},$$ where $$\frac1{p^*}=\frac1p-\frac{k}{n}$$ for the complete manifold(bounded geometry).