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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).

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