# Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in sup-norm).

(For simplicity we may just consider $M = \mathbb{R}^n$ with the usual Euclidean metric. Assuming bounded geometry implies that different possible definitions of Sobolev norms are equivalent.)

We equip $C_c^\infty(M)$, the space of all compactly supported smooth functions on $M$, with the countable family of Sobolev norms $\{\| \cdot \|_{W^{k,1}}\}_{k \ge 0}$ (it is important to me that we use $L^1$-integrability here). So we get a locally convex topological algebra which we still denote by $C_c^\infty(M)$.

Do we have $C_c^\infty(M) \ \hat{\otimes} \ C_c^\infty(N) \cong C_c^\infty(M \times N)$, where $\hat{\otimes}$ denotes the projective tensor product?

If we complete $C_c^\infty(M)$ in its family of norms, we get the infinite Sobolev space $W^{\infty,1}(M)$, which is a Frechet space.

Do we have $W^{\infty,1}(M) \ \hat{\otimes} \ W^{\infty,1}(N) \cong W^{\infty,1}(M \times N)$?

If the answers to the above questions are negative, what if we just use the four norms $\{W^{0,1}, W^{1,1}, W^{0, \infty}, W^{1, \infty}\}$ instead the countably many ones and ask the analogous two questions now? An affirmative answer here would be also good enough for my application though not very convenient.

The answer to the first question is no for a simple reason: I guess you do not want to complete the tensor product, but then then the left hand side is only dense in the right hand side.

For your second question, I think that the answer is yes, because $L^1(M)\hat\otimes L^1(N) \cong L^1(M\times N)$. But I have not yet seen a proof of this.

Maybe, the following paper which uses these (and other) spaces (but only on $\mathbb R^n$) might be helpful to you:

• Andreas Kriegl, Peter W. Michor, Armin Rainer: An exotic zoo of diffeomorphism groups on ℝn. 45 pages. arXiv:1404.7033. (pdf)

# Edit (twice):

With regard to you first question: If you the locally convex direct limit topology on $C^\infty(M)$ with respect to the embeddings of spaces $C^\infty_K(M)$ of smooth functions with support contained in a fixed compact $K$, then equality holds, since the projective tensor product respects direct limits in the category of locally convex spaces. Thus the algebraic tensor product is dense in the right hand side. Moreover, all tensor products between the projective and the injective coincide since the space is nuclear.

But you ask for another topology, the one induced by $W^{\infty,1}$. Density of the algebraic tensor product follows, since it is dense also in the finer topology.

This Frechet space is linearly isomorphic to $\ell^1\hat\otimes \mathcal s$ which is no longer nuclear, by a result of Dietmar Vogt, namely in the paper:

• MR0688001 Reviewed Vogt, Dietmar Sequence space representations of spaces of test functions and distributions. Functional analysis, holomorphy, and approximation theory (Rio de Janeiro, 1979), pp. 405–443, Lecture Notes in Pure and Appl. Math., 83, Dekker, New York, 1983. (Reviewer: M. Valdivia)

# 3rd Edit:

All your points with the exception of (iv) are correct. (iv) needs a proof. The space $W^{\infty,1}(\mathbb R^n)$ is called $\mathcal D_{L^1}$ in the book on Distributions of Laurent Schwartz. Density follows via the Stone Weierstrass theorem. The tensor product of the $C^\infty_c(M)$ spaces for the l.c. inductive limit topology is treated in:

• MR2296978 Reviewed Trèves, François Topological vector spaces, distributions and kernels. Unabridged republication of the 1967 original. Dover Publications, Inc., Mineola, NY, 2006. xvi+565 pp.

• The original version is in: MR0075539 Grothendieck, Alexandre Produits tensoriels topologiques et espaces nucléaires. (French) Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 pp. (Reviewer: J. Sebastião e Silva)

As for other tensor products, I think that the $\ell^p$-tensor product satisfies $W^{\infty,p}(M)\hat\otimes^p W^{\infty,p}(N) = W^{\infty,p}(M\times N)$. But this needs a proof, too.

Many different tensor products are described in:

• Peter W. Michor: Functors and categories of Banach spaces. Springer Lecture Notes 651, (1978), vi+99 pp.(pdf)
• If the left hand side is dense in the right hand side in my first question, doesn't this imply that the answer to my second question is "yes", because in my second question we just pass to the completions? Aug 29, 2014 at 7:31
• And could you please elaborate a bit on how to prove the denseness result in my first question? Thanks. Aug 29, 2014 at 7:32
• I will summarize your answer in order to make sure that I understood it correctly: (i) The topology on $C_c^\infty(M)$ that you talk about in the first paragraph of your edit is the "usual" one that one considers on this space (i.e., uniform convergence of all derivatives on all compacta). (ii) This topology is finer than the one that I consider (the one induced from $W^{\infty,1}(M)$). (iii) Since the inclusion $C_c^\infty(M) \otimes_{\mathrm{alg}} C_c^\infty(N) \hookrightarrow C_c^\infty(M \times N)$ is dense in the "usual" topology, it is also dense in "my" topology. Aug 30, 2014 at 12:57
• (iv) Passing to the completions, we may conclude that the answer to my second question is "yes", i.e., $W^{\infty,1}(M) \ \hat{\otimes} \ W^{\infty,1}(N) \cong W^{\infty,1}(M \times N)$. (v) The space $s$ you talk about is the same space $s$ as in this answer of you: mathoverflow.net/q/173588/13356. Aug 30, 2014 at 13:01
• Are my five points all correct? I also have two remaining questions: (1) What is a reference for the fact that $C_c^\infty(M) \otimes_{\mathrm{alg}} C_c^\infty(N) \hookrightarrow C_c^\infty(M \times N)$ is dense in the "usual" topology? This probably just boils down to a reference for the fact that the projective tensor product respects direct limits of locally convex spaces. (2) Since "my" topology does not make the spaces nuclear, what is the difference if we choose another tensor product in my question instead of the projective one? Thanks in advance and for the already given answer! Aug 30, 2014 at 13:07