Tensor product of certain Sobolev spaces on non-compact manifolds

Question

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.

Answer 1

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)