Kernel Identities and Vectorial Regularization

We present the method of"vectorial regularization"to prove kernel identities. This method is applied to derive both known kernel identities, e.g. $\dot{\mathcal{B}}_{xy}=\dot{\mathcal{B}}_x\widehat{\otimes}_\varepsilon\dot{\mathcal{B}}_y$, $\mathcal{D}'_{L^1,xy}=\mathcal{D}'_{L^1,x}\widehat{\otimes}_\pi\mathcal{D}'_{L^1,y}$, as well as new ones: $\dot{\mathcal{B}}'_{xy}=\dot{\mathcal{B}}'_x\widehat{\otimes}_\varepsilon\dot{\mathcal{B}}'_y$ and $\mathcal{D}_{L^1,xy}=\mathcal{D}_{L^1,x}\widehat{\otimes}_\pi\mathcal{D}_{L^1,y}$.


Introduction
In the following all distribution spaces are defined on the whole of R n , i.e., D ′ = D ′ (R n ), D ′ L p = D ′ L p (R n ), etc. A regularization property for a distribution T ∈ D ′ is a statement of the following form: Let 1 ≤ p ≤ ∞. For T ∈ D ′ the assertions 1. T ∈ D ′ L p 2. ∀ϕ ∈ D : ϕ * T ∈ L p are equivalent. By means of the "associated difference kernel" the equivalence above can be translated into the equivalence The "associated difference kernel" T (x − y) is defined in [10, pp. 103-104].
A vectorial regularization property reads as: Let E be a space of distributions and K(x, z) ∈ D ′ x (E z ). Then, wherein the ε-product is defined in [10, p. 18]. Note that by Corollaire 1 in [10, p. 47] The symbol ⊗ without subscript is used if ⊗ π = ⊗ ε , e.g., if one of the spaces is nuclear. If D ′ L p ⊗ π E is used instead of D ′ L p ⊗ ε E we speak of vectorial regularization with the completed projective tensor product.
By kernel identities, we understand statements as e.g. L. Schwartz's classical kernel theorem, i.e., Two fundamental examples of kernel identities are given in [5, chap. I, pp. 61, 90]: where X and Y are locally compact spaces. In order to abbreviate the notation, we will write these identities for X = R n and Y = R m as In [10], L. Schwartz found the algebraic and topological kernel identities Note that however, in Proposition 6 andḂ ′ xy =Ḃ ′ ⊗ εḂ ′ y (6) in Proposition 3 by vectorial regularization properties. Our proofs of the identities (2,3,5,6) show that they are all consequences of Grothendieck's fundamental examples (1). Also it turns out that in some cases the topological part of the kernel identities follows from the algebraic identity and abstract structural results, e.g. for complete spaces of distributions H the continuous embeddings impliy that the identity mapping H xy → H x ⊗ ε H y has a closed graph. In concrete cases, sequence-space representations can be used to check whether these spaces satisfy the assumptions of a suitable closed graph theorem. We use the notations of L. Schwartz in [12], e.g. the space of distributions E, E ′ , D, For vector-valued distributions, constant use is made of L. Schwartz' treatise [10,11]. Instead of K(x,ŷ) we simply write K(x, y) for kernels K(x, y) ∈ D ′ xy . Proposition 3 was presented in a talk given by the second author in Vienna, June, 2015.
2 Regularization and the injective tensor product Proposition 1. Let E be a complete space of distributions. For a kernel K(x, z) ∈ D ′ xz the following characterizations hold: Although for a normal space of distributions E the characterization 1 of this Proposition is a special case of Proposition 15 in [3], we include it nevertheless to keep the article self-contained.

Proof.
1. We first show the case of distributions.
is well-defined, linear and continuous according to Remarque 3 in [12, p. 202]. Hence also the mapping as by Proposition 1 in [10, p. 20] the ε-product of continuous linear mappings is again continuous.
From C 0,x ⊗ ε E ′ y ֒→Ḃ ′ xy and the invariance ofḂ ′ x,y under the coordinate transform Evaluation with e −|v| 2 ∈ S v yields Multiplication by e |w| 2 ∈ E w , which is possible by Theorem 7.1 in [9] leads to 2. The implication "⇒" is completely analogous to the case of distributions if we use that the convolution mapping * : E ′ ×Ḃ → C 0 is well defined and hypocontinuous since E ′ ֒→ D ′ L 1 andḂ ֒→ C 0 (see e.g. [7]). Let us show the implication "⇐". The vectorial scalar product of From this we deduce K(x, z) ∈Ḃ x ⊗ ε E z using the compatibility of the vector-valued scalar product with continuous linear mappings by [11, p. 18].
Remark 1. Note that it is possible to generalize this result to non-complete spaces of distributions but in this case the completed ε-tensor product has to be replaced by the ε-product.
Proposition 2 (see Proposition 17 in [10]). The space of smooth functions vanishing at infinity satisfies the kernel identityḂ xy =Ḃ x ⊗ εḂy algebraically and topologically.
Proof. In order to show the algebraic part, observe that for K(x, y) ∈ D ′ xy we get Let us now show the topological identity as well. As the ε-product of two continuous linear mappings is again continuous, we see that the mappinġ is continuous for all multi-indices α and β. Therefore the topology ofḂ x ⊗ εḂy is finer than the one ofḂ xy . Therefore these topologies are comparable Fréchet space topologies on the same vector space and hence they coincide. Proof. For K(x, y) ∈ D ′ xy we start with the characterization using the kernel theorem forḂ and D ′ as well as the commutativity of the ε-tensor product. From this we get from Proposition 1, x , which proves the algebraic part of the kernel identity. Using the sequence space repre-sentationḂ ′ = s ′ ⊗c 0 given in Theorem 3 in [2, p. 13] and we see by Proposition 7 in [2, p. 13] that bothḂ ′ xy andḂ ′ x ⊗Ḃ ′ y are complete ultrabornological (DF)-spaces. From the continuity of the embeddingṡ

Regularization and the projective tensor product
In order to proof a version of Proposition 1 for the projective tensor product, we need the following lemma. Lemma 1. For 1 < q < ∞ the following continuous embeddings hold: i.e., theses spaces are contained with a finer topology. Moreover these spaces are contained as dense subspaces.
Proof. From E xy = E x ⊗E y , we deduce that D L q ,x ⊗E y is a space of smooth functions. Using Lebesgue's theorem on dominated convergence we conclude that for f ∈ D L q ,xy the function R d x → D L q ,y , x → f (x, ·) has continuous derivatives of all order. Continuity of the embedding D L q ,xy ֒→ E x ⊗D L q ,y follows inductively from the Sobolev trace theorem, see, e.g., Theorem 5.36 in [1].
Given f ∈ S x ⊗D L q ,y , the inequality proves S x ⊗D L q ,y ֒→ D L q ,xy . The spaces are contained as dense subspaces since D xy ֒→ D x ⊗D y ⊂ S x ⊗D L q ,y and the injective tensor product preserves dense subspaces by Proposition 16.2.5 in [6, p. 349].
Proposition 4. Let E be a space of distributions and 1 ≤ p < ∞. For K(x, z) ∈ D ′ xz the following characterizations hold: is well-defined, linear and continuous according to [12, p. 204]. Hence also as the π-tensor product of continuous linear mappings is again a continuous and linear mapping.
⇐: Multiplication of K(x−y, z) ∈ D ′ y (L p x ⊗ π E z ) with δ(w−y) ∈ D y ⊗D ′ w according to Proposition 25 in [11, p. 120] yields Note that the inclusion L p ⊗E ′ ⊂ D ′ L p follows from L p ⊗E ′ ⊂ D ′ L p ⊗E ′ and froṁ B xy =Ḃ x ⊗ εḂy ֒→Ḃ x ⊗E y for p = 1 and from Lemma 1 and Multiplication by e |w| 2 ∈ E w according to Theorem 7.1 in [9, p. 31] yields K(u, z) ∈ D ′ w ⊗(D ′ L p ,u ⊗ π E z ) and hence K(u, z) ∈ D ′ L p ,u ⊗ π E z .
2. The implication "⇒" is completely analogous to the case of distributions if we use that the convolution mapping * : E ′ × D L p → L p is well defined and hypocontinuous since E ′ ֒→ D ′ L 1 and D L p ֒→ L p (see e.g. [7]). Let us show the implication "⇐". The vectorial scalar product of with ∂ α δ(y) ∈ E ′ y yields ∂ α x K(x, z) ∈ L p x ⊗ π E z for all α ∈ N n 0 . From this we deduce K(x, z) ∈ D L p ,x ⊗ π E z using the compatibility of the vector-valued scalar product with continuous linear mappings by [11, p. 18].
Remark 2. More general, the proof of equivalence 1 in Proposition 4 also works in the following situation. Let H ′ be a space of distributions and K a space of functions such that the convolution mapping H ′ × D → K is hypocontinuous. If additionally the embeddings are well-defined and continuous, for kernels K(x, y) ∈ D ′ x,y we get the following equivalence Examples of spaces H ′ satisfying condition (7) are duals of normal spaces of distributions H where the embeddings S x ⊗H y ֒→ H x,y ֒→ E x ⊗H y are well-defined and continuous. Note that the spaces S ⊗H and E ⊗H are spaces of H-valued smooth functions. We refer to [8] for a detailed treatment of these spaces.
In the following we will discuss two kernel-identities as applications of Proposition 4.
Proposition 5 (see Proposition 38 in [10, p. 135]). The space of integrable distributions satisfies the kernel identity algebraically and topologically.
Proof. For K(x, y) ∈ D ′ x,y we have the equivalence x,y which follows from the characterization of D ′ L 1 by regularization given in Théorème XXV in [12, p. 201]. Using the kernel identities D ′ x,y = D ′ x ⊗D ′ y and L 1 xy = L 1 x ⊗ π L 1 y , we obtain Applying Proposition 4 twice to the line above, we finally get i.e. we have shown the algebraic identity D ′ L 1 ,xy = D ′ L 1 ,x ⊗ π D ′ L 1 ,y . In order to prove the continuity of the identity mapping it is sufficient to show the continuity of the bilinear mapping The continuity of this mapping follows from the separate continuity due to the fact that for (DF)-spaces separate continuity of bilinear maps implies continuity. The separate continuity follows immediately by the closed graph theorem. By de Wilde's closed graph theorem (Theorem 5.4.1 in [6, p. 92]) the identity is a topological isomorphism because D ′ L 1 ,xy is ultrabornological and D ′ L 1 ,x ⊗ π D ′ L 1 ,y is a complete (DF)-space and, hence, has a completing web by Proposition 12.4.6 in [6, p. 260]. Proposition 6. The space of integrable smooth functions satisfies the kernel identity D L 1 ,xy = D L 1 ,x ⊗ π D L 1 ,y algebraically and toplogically.
Proof. For S ∈ D ′ we get S ∈ D L 1 ⇔ S(x − y) ∈ E y ⊗L 1 x and therefore for K ∈ D ′ xy , K(x, y) ∈ D L 1 ,xy ⇔ K(x − z, y − w) ∈ E zw ⊗L 1 xy .
From this equivalence, we deduce K(x, y) ∈ D L 1 ,xy ⇔ K(x − z, y − w) ∈ E z ⊗E w ⊗ L 1 x ⊗ π L 1 y = E z ⊗L 1 x ⊗ π E w ⊗L 1 y using the classical kernel identities E xy = E x ⊗E y and L 1 xy = L 1 x ⊗ π L 1 y . From Proposition 4, applied twice, we get K(x, y) ∈ D L 1 ,xy ⇔ K(x, y − w) ∈ D L 1 ,x ⊗ π E w ⊗L 1 y ⇔ K(x, y) ∈ D L 1 ,xy .
As the π-tensor product of continuous mappings is continuous, the mapping is continuous for all multi-indices α and β. Hence the π-topology is finer than the topology of D L 1 . As these topologies are comparable Fréchet space topologies on the same vector space they coincide by the closed graph theorem.