The Schwartz Kernel Theorem and Distributions of Finite Order

The Schwartz kernel theorem and distributions of ﬁnite order

Nenad Antoni´c

Department of Mathematics Faculty of Science University of Zagreb

A conference in memory of Todor Gramchev Torino, 1st February 2017

Joint work with Marko Erceg and Marin Miˇsur Distributions of anisotropic order H-distributions Functions of anisotropic smoothness Deﬁnition and tensor products Conjectures

Schwartz kernel theorem Statement and strategies The proof Consequence for H-distributions

2 Theorem. [H¨ormander-Mihlin] Let ψ ∈ L∞(Rd) have partial derivatives of d order less than or equal to κ = [ 2 ] + 1. If for some k > 0 Z d α 2 2 d−2|α| (∀r > 0)(∀α ∈ N0) |α| 6 κ =⇒ |∂ ψ(ξ)| dξ 6 k r , r 2 6|ξ|6r

then for any p ∈ h1, ∞i and the associated multiplier operator Aψ there exists a Cd (depending only on the dimension d) such that 1 kA k p p C max p, (k + kψk ) . ψ L →L 6 d p − 1 ∞

κ d−1 d For ψ ∈ C (S ), extended by homogeneity to R∗, we can take k = kψkCκ .

H¨ormander-Mihlintheorem ψ : Rd → C is a Fourier multiplier on Lp(Rd) if

F¯(ψF(θ)) ∈ Lp(Rd) , for θ ∈ S(Rd),

and S(Rd) 3 θ 7→ F¯(ψF(θ)) ∈ Lp(Rd) p d p d can be extended to a continuous mapping Aψ :L (R ) → L (R ).

3 κ d−1 d For ψ ∈ C (S ), extended by homogeneity to R∗, we can take k = kψkCκ .

H¨ormander-Mihlintheorem ψ : Rd → C is a Fourier multiplier on Lp(Rd) if

F¯(ψF(θ)) ∈ Lp(Rd) , for θ ∈ S(Rd),

and S(Rd) 3 θ 7→ F¯(ψF(θ)) ∈ Lp(Rd) p d p d can be extended to a continuous mapping Aψ :L (R ) → L (R ). Theorem. [H¨ormander-Mihlin] Let ψ ∈ L∞(Rd) have partial derivatives of d order less than or equal to κ = [ 2 ] + 1. If for some k > 0 Z d α 2 2 d−2|α| (∀r > 0)(∀α ∈ N0) |α| 6 κ =⇒ |∂ ψ(ξ)| dξ 6 k r , r 2 6|ξ|6r

then for any p ∈ h1, ∞i and the associated multiplier operator Aψ there exists a Cd (depending only on the dimension d) such that 1 kA k p p C max p, (k + kψk ) . ψ L →L 6 d p − 1 ∞

3 H¨ormander-Mihlintheorem ψ : Rd → C is a Fourier multiplier on Lp(Rd) if

F¯(ψF(θ)) ∈ Lp(Rd) , for θ ∈ S(Rd),

then for any p ∈ h1, ∞i and the associated multiplier operator Aψ there exists a Cd (depending only on the dimension d) such that 1 kA k p p C max p, (k + kψk ) . ψ L →L 6 d p − 1 ∞

κ d−1 d For ψ ∈ C (S ), extended by homogeneity to R∗, we can take k = kψkCκ .

3 µ is the H-distribution corresponding to (a subsequence of) (un) and (vn). d d If (un), (vn) are deﬁned on Ω ⊆ R , extension by zero to R preserves the convergence, and we can apply the Theorem. µ is supported on Cl Ω × Sd−1. p d q d 0 We distinguish un ∈ L (R ) and vn ∈ L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn). 2 d The assumptions imply un, vn −* 0 in Lloc(R ), resulting in a distribution µ of order zero (an unbounded Radon measure, not a general distribution). The novelty in Theorem is for p < 2. p d k q d l For vector-valued un ∈ L (R ; C ) and vn ∈ L (R ; C ), the result is a matrix valued distribution µ = [µij ], i ∈ 1..k and j ∈ 1..l. The H-distribution would correspond to a non-diagonal block for an H-measure.

Existence of H-distributions

p d Theorem. [N.A. & D. Mitrovi´c(2011)] If un −* 0 in L (R ) and ∗ q d 0 vn −−* v in L (R ) for some q > max{p , 2}, then there exist subsequences 0 d d−1 (un0 ), (vn0 ) and a complex valued distribution µ ∈ D (R × S ), such that ∞ d κ d−1 for every ϕ1, ϕ2 ∈ Cc (R ) and ψ ∈ C (S ) we have: Z Z lim Aψ(ϕ1un0 )(x)(ϕ2vn0 )(x)dx = lim (ϕ1un0 )(x)A (ϕ2vn0 )(x)dx 0 0 ψ n Rd n Rd

= hµ, ϕ1ϕ2ψi .

4 d d If (un), (vn) are deﬁned on Ω ⊆ R , extension by zero to R preserves the convergence, and we can apply the Theorem. µ is supported on Cl Ω × Sd−1. p d q d 0 We distinguish un ∈ L (R ) and vn ∈ L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn). 2 d The assumptions imply un, vn −* 0 in Lloc(R ), resulting in a distribution µ of order zero (an unbounded Radon measure, not a general distribution). The novelty in Theorem is for p < 2. p d k q d l For vector-valued un ∈ L (R ; C ) and vn ∈ L (R ; C ), the result is a matrix valued distribution µ = [µij ], i ∈ 1..k and j ∈ 1..l. The H-distribution would correspond to a non-diagonal block for an H-measure.

Existence of H-distributions

= hµ, ϕ1ϕ2ψi .

µ is the H-distribution corresponding to (a subsequence of) (un) and (vn).

4 p d q d 0 We distinguish un ∈ L (R ) and vn ∈ L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn). 2 d The assumptions imply un, vn −* 0 in Lloc(R ), resulting in a distribution µ of order zero (an unbounded Radon measure, not a general distribution). The novelty in Theorem is for p < 2. p d k q d l For vector-valued un ∈ L (R ; C ) and vn ∈ L (R ; C ), the result is a matrix valued distribution µ = [µij ], i ∈ 1..k and j ∈ 1..l. The H-distribution would correspond to a non-diagonal block for an H-measure.

Existence of H-distributions

= hµ, ϕ1ϕ2ψi .

4 p d k q d l For vector-valued un ∈ L (R ; C ) and vn ∈ L (R ; C ), the result is a matrix valued distribution µ = [µij ], i ∈ 1..k and j ∈ 1..l. The H-distribution would correspond to a non-diagonal block for an H-measure.

Existence of H-distributions

= hµ, ϕ1ϕ2ψi .

4 The H-distribution would correspond to a non-diagonal block for an H-measure.

Existence of H-distributions

= hµ, ϕ1ϕ2ψi .

µ is the H-distribution corresponding to (a subsequence of) (un) and (vn). d d If (un), (vn) are deﬁned on Ω ⊆ R , extension by zero to R preserves the convergence, and we can apply the Theorem. µ is supported on Cl Ω × Sd−1. p d q d 0 We distinguish un ∈ L (R ) and vn ∈ L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn). 2 d The assumptions imply un, vn −* 0 in Lloc(R ), resulting in a distribution µ of order zero (an unbounded Radon measure, not a general distribution). The novelty in Theorem is for p < 2. p d k q d l For vector-valued un ∈ L (R ; C ) and vn ∈ L (R ; C ), the result is a matrix valued distribution µ = [µij ], i ∈ 1..k and j ∈ 1..l.

4 Existence of H-distributions

= hµ, ϕ1ϕ2ψi .

p/p0 kΦ (u)k 0 kuk . p Lp (Rd) 6 Lp(Rd)

p d p0 d It maps bounded sets in Lloc(R ) topology to bounded sets in Lloc(R ) p topology. Hence for an L bounded sequence (un), we get that (Φp(un)) is p0 d weakly precompact in Lloc(R ). p d p0 d It is continuous from Lloc(R ) to Lloc(R ).

A particular Nemycki˘ıoperator

0 Canonical choice of Lp sequence corresponding to an Lp, p ∈ h1, ∞i, sequence p d (un) is given by vn = Φp(un), where Φp is an operator from L (R ) to p0 d p−2 L (R ) deﬁned by Φp(u) = |u| u.

5 p d p0 d It maps bounded sets in Lloc(R ) topology to bounded sets in Lloc(R ) p topology. Hence for an L bounded sequence (un), we get that (Φp(un)) is p0 d weakly precompact in Lloc(R ). p d p0 d It is continuous from Lloc(R ) to Lloc(R ).

A particular Nemycki˘ıoperator

0 Canonical choice of Lp sequence corresponding to an Lp, p ∈ h1, ∞i, sequence p d (un) is given by vn = Φp(un), where Φp is an operator from L (R ) to p0 d p−2 L (R ) deﬁned by Φp(u) = |u| u. p d p0 d Φp is a nonlinear Nemytski˘ıoperator, continuous from L (R ) to L (R ) and additionally we have the following bound

p/p0 kΦ (u)k 0 kuk . p Lp (Rd) 6 Lp(Rd)

5 p d p0 d It is continuous from Lloc(R ) to Lloc(R ).

A particular Nemycki˘ıoperator

p/p0 kΦ (u)k 0 kuk . p Lp (Rd) 6 Lp(Rd)

p d p0 d It maps bounded sets in Lloc(R ) topology to bounded sets in Lloc(R ) p topology. Hence for an L bounded sequence (un), we get that (Φp(un)) is p0 d weakly precompact in Lloc(R ).

5 A particular Nemycki˘ıoperator

p/p0 kΦ (u)k 0 kuk . p Lp (Rd) 6 Lp(Rd)

5 p d Simple change of variables: kunkLp(Rd) = kukLp(Rd) and un −* 0 in L (R ). d Indeed, the sequence is bounded, while for ϕ ∈ Cc(R ) Z Z d/p un(x)ϕ(x)dx = n u(n(x − z))ϕ(x)dx Rd Rd Z = nd/p−du(y)ϕ(y/n + z)dy Rd 1 Z = d/p0 u(y)χsupp u(y)ϕ(y/n + z)dy n Rd vol(supp u) 1/p0 6 d kukLp(Rd) max |ϕ|. n Rd Passing to the limit, we get our claim.

Actually, the H-distribution corresponding to sequences (un) and (Φp(un)) is κ d−1 given by δz ν, where ν is a distribution on C (S ) deﬁned for ψ ∈ Cκ(Sd−1) by Z p−2 hν, ψi = u(x)Aψ¯(|u| u)(x)dx. Rd

Example: concentration d p d p d u ∈ Lc (R ), and deﬁne un(x) = n u(n(x − z)) for some z ∈ R .

6 d Indeed, the sequence is bounded, while for ϕ ∈ Cc(R ) Z Z d/p un(x)ϕ(x)dx = n u(n(x − z))ϕ(x)dx Rd Rd Z = nd/p−du(y)ϕ(y/n + z)dy Rd 1 Z = d/p0 u(y)χsupp u(y)ϕ(y/n + z)dy n Rd vol(supp u) 1/p0 6 d kukLp(Rd) max |ϕ|. n Rd Passing to the limit, we get our claim.

Example: concentration d p d p d u ∈ Lc (R ), and deﬁne un(x) = n u(n(x − z)) for some z ∈ R . p d Simple change of variables: kunkLp(Rd) = kukLp(Rd) and un −* 0 in L (R ).

6 Actually, the H-distribution corresponding to sequences (un) and (Φp(un)) is κ d−1 given by δz ν, where ν is a distribution on C (S ) deﬁned for ψ ∈ Cκ(Sd−1) by Z p−2 hν, ψi = u(x)Aψ¯(|u| u)(x)dx. Rd

Example: concentration d p d p d u ∈ Lc (R ), and deﬁne un(x) = n u(n(x − z)) for some z ∈ R . p d Simple change of variables: kunkLp(Rd) = kukLp(Rd) and un −* 0 in L (R ). d Indeed, the sequence is bounded, while for ϕ ∈ Cc(R ) Z Z d/p un(x)ϕ(x)dx = n u(n(x − z))ϕ(x)dx Rd Rd Z = nd/p−du(y)ϕ(y/n + z)dy Rd 1 Z = d/p0 u(y)χsupp u(y)ϕ(y/n + z)dy n Rd vol(supp u) 1/p0 6 d kukLp(Rd) max |ϕ|. n Rd Passing to the limit, we get our claim.

6 Example: concentration d p d p d u ∈ Lc (R ), and deﬁne un(x) = n u(n(x − z)) for some z ∈ R . p d Simple change of variables: kunkLp(Rd) = kukLp(Rd) and un −* 0 in L (R ). d Indeed, the sequence is bounded, while for ϕ ∈ Cc(R ) Z Z d/p un(x)ϕ(x)dx = n u(n(x − z))ϕ(x)dx Rd Rd Z = nd/p−du(y)ϕ(y/n + z)dy Rd 1 Z = d/p0 u(y)χsupp u(y)ϕ(y/n + z)dy n Rd vol(supp u) 1/p0 6 d kukLp(Rd) max |ϕ|. n Rd Passing to the limit, we get our claim.

6 By Cl,m(Ω) we denote the space of functions f on Ω, such that for any d r α ∈ N0 and β ∈ N0, if |α| 6 l and |β| 6 m, α,β α β ∂ f = ∂x ∂y f ∈ C(Ω) . Cl,m(Ω) becomes a Fr´echetspace if we deﬁne a sequence of seminorms

l,m α,β p (f) := max k∂ fk ∞ , Kn L (Kn) |α|6l,|β|6m

where Kn ⊆ Ω are compacts, such that Ω = ∪n∈NKn and Kn ⊆ IntKn+1. For a compact set K ⊆ Ω we deﬁne a subspace of Cl,m(Ω)

l,m n l,m o CK (Ω) := f ∈ C (Ω) : supp f ⊆ K .

This subspace inherits the topology from Cl,m(Ω), which is, when considered only on the subspace, a norm topology determined by

l,m kfkl,m,K := pK (f) , l,m and CK (Ω) is a Banach space (it can be identified with a proper subspace of Cl,m(K)). However, if m = ∞ (or l = ∞), then we shall not get a Banach space, but a Fréchetspace. As in the isotropic case, an increasing sequence of seminorms that makes Cl,∞(Ω) a Fréchetspace is given by (pl,k ), k ∈ N . Kn Kn 0

Functions of anisotropic smoothness Let X and Y be open sets in Rd and Rr (or C∞ manifolds), Ω ⊆ X × Y .

7 Cl,m(Ω) becomes a Fr´echetspace if we deﬁne a sequence of seminorms

l,m α,β p (f) := max k∂ fk ∞ , Kn L (Kn) |α|6l,|β|6m

where Kn ⊆ Ω are compacts, such that Ω = ∪n∈NKn and Kn ⊆ IntKn+1. For a compact set K ⊆ Ω we deﬁne a subspace of Cl,m(Ω)

l,m n l,m o CK (Ω) := f ∈ C (Ω) : supp f ⊆ K .

This subspace inherits the topology from Cl,m(Ω), which is, when considered only on the subspace, a norm topology determined by

Functions of anisotropic smoothness Let X and Y be open sets in Rd and Rr (or C∞ manifolds), Ω ⊆ X × Y . By Cl,m(Ω) we denote the space of functions f on Ω, such that for any d r α ∈ N0 and β ∈ N0, if |α| 6 l and |β| 6 m, α,β α β ∂ f = ∂x ∂y f ∈ C(Ω) .

7 For a compact set K ⊆ Ω we deﬁne a subspace of Cl,m(Ω)

l,m n l,m o CK (Ω) := f ∈ C (Ω) : supp f ⊆ K .

This subspace inherits the topology from Cl,m(Ω), which is, when considered only on the subspace, a norm topology determined by

l,m α,β p (f) := max k∂ fk ∞ , Kn L (Kn) |α|6l,|β|6m

where Kn ⊆ Ω are compacts, such that Ω = ∪n∈NKn and Kn ⊆ IntKn+1.

7 Functions of anisotropic smoothness Let X and Y be open sets in Rd and Rr (or C∞ manifolds), Ω ⊆ X × Y . By Cl,m(Ω) we denote the space of functions f on Ω, such that for any d r α ∈ N0 and β ∈ N0, if |α| 6 l and |β| 6 m, α,β α β ∂ f = ∂x ∂y f ∈ C(Ω) . Cl,m(Ω) becomes a Fr´echetspace if we deﬁne a sequence of seminorms

l,m α,β p (f) := max k∂ fk ∞ , Kn L (Kn) |α|6l,|β|6m

where Kn ⊆ Ω are compacts, such that Ω = ∪n∈NKn and Kn ⊆ IntKn+1. For a compact set K ⊆ Ω we deﬁne a subspace of Cl,m(Ω)

l,m n l,m o CK (Ω) := f ∈ C (Ω) : supp f ⊆ K .

This subspace inherits the topology from Cl,m(Ω), which is, when considered only on the subspace, a norm topology determined by

Cl,m(Ω) ,→ Cl,m (Ω) , Kn Kn+1

the inclusion being continuous. Also, the topology induced on Cl,m(Ω) by that Kn of Cl,m (Ω) coincides with the original one, and Cl,m(Ω) (as a Banach space Kn+1 Kn in that topology) is a closed subspace of Cl,m (Ω). Then we have that the Kn+1 inductive limit topology on Cl,m(Ω) induces on each Cl,m(Ω) the original c Kn l,m topology, while a subset of Cc (Ω) is bounded if and only if it is contained in one Cl,m(Ω), and bounded there. Cl,m(Ω) is a barelled space. Kn c ∞ l,m Of course, Cc (Ω) ,→ Cc (Ω) is a continuous and dense imbedding.

Functions of anisotropic smoothness (cont.)

We can also consider the space

l,m [ l,m C (Ω) := C (Ω) , c Kn n∈N

of all functions with compact support in Cl,m(Ω), and equip it by a stronger topology than the one induced from Cl,m(Ω): by the topology of strict inductive limit.

8 Also, the topology induced on Cl,m(Ω) by that Kn of Cl,m (Ω) coincides with the original one, and Cl,m(Ω) (as a Banach space Kn+1 Kn in that topology) is a closed subspace of Cl,m (Ω). Then we have that the Kn+1 inductive limit topology on Cl,m(Ω) induces on each Cl,m(Ω) the original c Kn l,m topology, while a subset of Cc (Ω) is bounded if and only if it is contained in one Cl,m(Ω), and bounded there. Cl,m(Ω) is a barelled space. Kn c ∞ l,m Of course, Cc (Ω) ,→ Cc (Ω) is a continuous and dense imbedding.

Functions of anisotropic smoothness (cont.)

We can also consider the space

l,m [ l,m C (Ω) := C (Ω) , c Kn n∈N

of all functions with compact support in Cl,m(Ω), and equip it by a stronger topology than the one induced from Cl,m(Ω): by the topology of strict inductive limit. More precisely, it can easily be checked that

Cl,m(Ω) ,→ Cl,m (Ω) , Kn Kn+1

the inclusion being continuous.

8 ∞ l,m Of course, Cc (Ω) ,→ Cc (Ω) is a continuous and dense imbedding.

Functions of anisotropic smoothness (cont.)

We can also consider the space

l,m [ l,m C (Ω) := C (Ω) , c Kn n∈N

Cl,m(Ω) ,→ Cl,m (Ω) , Kn Kn+1

8 Functions of anisotropic smoothness (cont.)

We can also consider the space

l,m [ l,m C (Ω) := C (Ω) , c Kn n∈N

Cl,m(Ω) ,→ Cl,m (Ω) , Kn Kn+1

8 ∞ l,m 0 Clearly, Cc (Ω) ,→ Cc (Ω) ,→ D (Ω), with continuous and dense imbeddings, l,m 0 thus Cc (Ω) is a normal space of distributions, hence its dual Dl,m(Ω) forms a subspace of D0(Ω). If we equip it with a strong topology, it is even continuously imbedded in D0(Ω).

Lemma. Let X and Y be C∞ manifolds. For a linear functional u on l,m Cc (X × Y ), the following statements are equivalent 0 a) u ∈ Dl,m(X × Y ), l,m l,m b) (∀K ∈ K(X × Y ))(∃C > 0)(∀Ψ ∈ CK (X × Y )) |hu, Ψi| 6 CpK (Ψ). Statement (b) of previous lemma implies:

l m (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) . The reverse implication would have signiﬁcantly greater practical use.

Distributions of anisotropic order

9 Lemma. Let X and Y be C∞ manifolds. For a linear functional u on l,m Cc (X × Y ), the following statements are equivalent 0 a) u ∈ Dl,m(X × Y ), l,m l,m b) (∀K ∈ K(X × Y ))(∃C > 0)(∀Ψ ∈ CK (X × Y )) |hu, Ψi| 6 CpK (Ψ). Statement (b) of previous lemma implies:

l m (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) . The reverse implication would have signiﬁcantly greater practical use.

Distributions of anisotropic order

Deﬁnition. A distribution of order l in x and order m in y is any linear l,m functional on Cc (Ω), continuous in the strict inductive limit topology. We 0 denote the space of such functionals by Dl,m(Ω). ∞ l,m 0 Clearly, Cc (Ω) ,→ Cc (Ω) ,→ D (Ω), with continuous and dense imbeddings, l,m 0 thus Cc (Ω) is a normal space of distributions, hence its dual Dl,m(Ω) forms a subspace of D0(Ω). If we equip it with a strong topology, it is even continuously imbedded in D0(Ω).

9 Statement (b) of previous lemma implies:

l m (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) . The reverse implication would have signiﬁcantly greater practical use.

Distributions of anisotropic order

9 Distributions of anisotropic order

l m (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) . The reverse implication would have signiﬁcantly greater practical use.

9 ∞ 0 0 Theorem. Let X and Y be C manifolds, u ∈ Dl(X) and v ∈ Dm(Y ). Then

0 l m ∃!w ∈ Dl,m(X×Y ) ∀ϕ ∈ Cc(X) ∀ψ ∈ Cc (Y ) hw, ϕψi = hu, ϕihv, ψi.

l,m Furthermore, for any Φ ∈ Cc (X × Y ), function V : x 7→ hv, Φ(x, ·)i is in l m Cc(X), while U : y 7→ hu, Φ(·, y)i is in Cc (Y ), and we have that hw, Φi = hu, V i = hv, Ui.

Tensor product of distributions

0 In order to better understand the properties of elements of Dl,m(Ω), we shall relate them to tensor products. l m The ﬁrst step is to consider the algebraic tensor product Cc(X) Cc (Y ), the vector space of all (ﬁnite) linear combinations of functions of the form l,m (φ ψ)(x, y) := φ(x)ψ(y). This is a vector subspace of Cc (X × Y ).

10 Tensor product of distributions

∞ 0 0 Theorem. Let X and Y be C manifolds, u ∈ Dl(X) and v ∈ Dm(Y ). Then

0 l m ∃!w ∈ Dl,m(X×Y ) ∀ϕ ∈ Cc(X) ∀ψ ∈ Cc (Y ) hw, ϕψi = hu, ϕihv, ψi.

l,m Furthermore, for any Φ ∈ Cc (X × Y ), function V : x 7→ hv, Φ(x, ·)i is in l m Cc(X), while U : y 7→ hu, Φ(·, y)i is in Cc (Y ), and we have that hw, Φi = hu, V i = hv, Ui.

10 Simple operations

0 l,m Lemma. If u ∈ Dl,m(X × Y ) then, for any ψ ∈ C (X × Y ), ψu is a well deﬁned distribution of order at most (l, m).

0 Theorem. Let u ∈ Dl,m(X × Y ) and take F ⊆ X × Y relatively compact set such that supp u ⊆ F . Then there exists unique linear functional u˜ on l,m Q := {ϕ ∈ C (X × Y ): F ∩ supp ϕ b X × Y } such that l,m a) (∀ϕ ∈ Cc (X × Y )) hu,˜ ϕi = hu, ϕi, b) (∀ϕ ∈ Cl,m(X × Y )) F ∩ supp ϕ = ∅ =⇒ hu,˜ ϕi = 0. The domain of u˜ is largest for F = supp u.

11 0 d d−1 If it were true, then the H-distribution µ would belong to D0,κ(R × S ), i.e. it would be a distribution of order 0 in x and of order not more than κ in ξ. Indeed, from the proof of the existence theorem, we already have 0 d d−1 µ ∈ D (R × S ) and the following bound with ϕ := ϕ1ϕ2:

|hµ, ϕ ψi| CkψkCκ(Sd−1)kϕkC (Rd) , 6 Kl where C does not depend on ϕ and ψ. Most likely it is not true! We need a more complicated result.

First conjecture

∞ l,m Let X,Y be C manifolds and u a linear functional on Cc (X × Y ). If u ∈ D0(X × Y ) and satisﬁes

∞ ∞ (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) ,

0 then u can be uniquely extended to Dl,m(X × Y ).

12 Indeed, from the proof of the existence theorem, we already have 0 d d−1 µ ∈ D (R × S ) and the following bound with ϕ := ϕ1ϕ2:

|hµ, ϕ ψi| CkψkCκ(Sd−1)kϕkC (Rd) , 6 Kl where C does not depend on ϕ and ψ. Most likely it is not true! We need a more complicated result.

First conjecture

∞ l,m Let X,Y be C manifolds and u a linear functional on Cc (X × Y ). If u ∈ D0(X × Y ) and satisﬁes

∞ ∞ (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) ,

12 We need a more complicated result.

First conjecture

∞ l,m Let X,Y be C manifolds and u a linear functional on Cc (X × Y ). If u ∈ D0(X × Y ) and satisﬁes

∞ ∞ (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) ,

0 then u can be uniquely extended to Dl,m(X × Y ). 0 d d−1 If it were true, then the H-distribution µ would belong to D0,κ(R × S ), i.e. it would be a distribution of order 0 in x and of order not more than κ in ξ. Indeed, from the proof of the existence theorem, we already have 0 d d−1 µ ∈ D (R × S ) and the following bound with ϕ := ϕ1ϕ2:

|hµ, ϕ ψi| CkψkCκ(Sd−1)kϕkC (Rd) , 6 Kl where C does not depend on ϕ and ψ. Most likely it is not true!

12 First conjecture

∞ l,m Let X,Y be C manifolds and u a linear functional on Cc (X × Y ). If u ∈ D0(X × Y ) and satisﬁes

∞ ∞ (∀K ∈ K(X))(∀L ∈ K(Y )(∃C > 0)(∀ϕ ∈ CK (X))(∀ψ ∈ CL (Y )) l m |hu, ϕ ψi| 6 CpK (ϕ)pL (ψ) ,

|hµ, ϕ ψi| CkψkCκ(Sd−1)kϕkC (Rd) , 6 Kl where C does not depend on ϕ and ψ. Most likely it is not true! We need a more complicated result.

12 Distributions of anisotropic order H-distributions Functions of anisotropic smoothness Deﬁnition and tensor products Conjectures

Schwartz kernel theorem Statement and strategies The proof Consequence for H-distributions

13 Schwartz kernel theorem

Theorem. Let X and Y be two diﬀerentiable manifolds. 0 l a) Let K ∈ Dl,m(X × Y ). Then for each ϕ ∈ Cc(X) the linear form Kϕ, deﬁned by ψ 7→ hK, ϕ ψi, is a distribution of order not more than m on Y . l Furthermore, the mapping ϕ 7→ Kϕ, taking Cc(X) with its inductive limit 0 topology to Dm(Y ) with weak ∗ topology, is linear and continuous. l 0 b) Let A :Cc(X) → Dm(Y ) be a continuous linear operator, in the pair of topologies as above. Then there exists unique distribution K ∈ D0(X × Y ) ∞ ∞ such that for any ϕ ∈ Cc (X) and ψ ∈ Cc (Y )

hK, ϕψi = hKϕ, ψi = hAϕ, ψi. 0 Furthermore, K ∈ Dl,d(m+2)(X × Y ).

14 ◦ constructive proof? (Simanca, Gask, Ehrenpreis) ◦ nuclear spaces? (Tr`eves) ◦ structure theorem, on manifolds (Dieudonne)

Diﬀerent strategies of proof

◦ regularisation? (Schwartz)

15 ◦ nuclear spaces? (Tr`eves) ◦ structure theorem, on manifolds (Dieudonne)

Diﬀerent strategies of proof

◦ regularisation? (Schwartz) ◦ constructive proof? (Simanca, Gask, Ehrenpreis)

15 ◦ structure theorem, on manifolds (Dieudonne)

Diﬀerent strategies of proof

◦ regularisation? (Schwartz) ◦ constructive proof? (Simanca, Gask, Ehrenpreis) ◦ nuclear spaces? (Tr`eves)

15 Diﬀerent strategies of proof

◦ regularisation? (Schwartz) ◦ constructive proof? (Simanca, Gask, Ehrenpreis) ◦ nuclear spaces? (Tr`eves) ◦ structure theorem, on manifolds (Dieudonne)

15 m i.e. for H ∈ K(Y ), mapping ψ 7→ hKϕ, ψi is a cont. lin. funct. on CH (Y ). We can assume X and Y to be open subsets of Rd and Rr. Indeed, first take an open covering of Y , consisting of chart domains, and a partition of unity (fα) subordinate to that covering such that P α fα(y) = 1, y ∈ H (note that the sum is finite). Similarly for ϕ, thus limiting ourselves to domains of a pair of charts. By [Gösser,Kunzinger & al., Chapter 3.1.4], we can identify distributions localised on chart domains with distributions on subsets of Rd and Rr. Thus, in what follows we shall assume that X and Y are open subsets of Rd and Rr. We shall therefore show that there exists a constant C > 0 such that for any m ψ ∈ CH (Y ) it holds

β |hKϕ, ψi| 6 C max k∂ ψkL∞(H) , |β|6m for m ﬁnite, while for m = ∞ we should modify the above to

0 ∞ β (∃ m ∈ N)(∃ C > 0)(∀ ψ ∈ CH (Y )) |hKϕ, ψi| C max k∂ ψkL∞(H) . 6 0 |β|6m

From kernel to operator (a) l m ϕ ∈ Cc(X); prove the continuity of Kϕ on Cc (Y ) (it is clearly linear since the tensor product is bilinear, while K is linear).

16 We can assume X and Y to be open subsets of Rd and Rr. Indeed, first take an open covering of Y , consisting of chart domains, and a partition of unity (fα) subordinate to that covering such that P α fα(y) = 1, y ∈ H (note that the sum is finite). Similarly for ϕ, thus limiting ourselves to domains of a pair of charts. By [Gösser,Kunzinger & al., Chapter 3.1.4], we can identify distributions localised on chart domains with distributions on subsets of Rd and Rr. Thus, in what follows we shall assume that X and Y are open subsets of Rd and Rr. We shall therefore show that there exists a constant C > 0 such that for any m ψ ∈ CH (Y ) it holds

β |hKϕ, ψi| 6 C max k∂ ψkL∞(H) , |β|6m for m ﬁnite, while for m = ∞ we should modify the above to

0 ∞ β (∃ m ∈ N)(∃ C > 0)(∀ ψ ∈ CH (Y )) |hKϕ, ψi| C max k∂ ψkL∞(H) . 6 0 |β|6m

16 By [G¨osser,Kunzinger & al., Chapter 3.1.4], we can identify distributions localised on chart domains with distributions on subsets of Rd and Rr. Thus, in what follows we shall assume that X and Y are open subsets of Rd and Rr. We shall therefore show that there exists a constant C > 0 such that for any m ψ ∈ CH (Y ) it holds

β |hKϕ, ψi| 6 C max k∂ ψkL∞(H) , |β|6m for m ﬁnite, while for m = ∞ we should modify the above to

0 ∞ β (∃ m ∈ N)(∃ C > 0)(∀ ψ ∈ CH (Y )) |hKϕ, ψi| C max k∂ ψkL∞(H) . 6 0 |β|6m

16 We shall therefore show that there exists a constant C > 0 such that for any m ψ ∈ CH (Y ) it holds

β |hKϕ, ψi| 6 C max k∂ ψkL∞(H) , |β|6m for m ﬁnite, while for m = ∞ we should modify the above to

0 ∞ β (∃ m ∈ N)(∃ C > 0)(∀ ψ ∈ CH (Y )) |hKϕ, ψi| C max k∂ ψkL∞(H) . 6 0 |β|6m

From kernel to operator (a) l m ϕ ∈ Cc(X); prove the continuity of Kϕ on Cc (Y ) (it is clearly linear since the tensor product is bilinear, while K is linear). m i.e. for H ∈ K(Y ), mapping ψ 7→ hKϕ, ψi is a cont. lin. funct. on CH (Y ). We can assume X and Y to be open subsets of Rd and Rr. Indeed, first take an open covering of Y , consisting of chart domains, and a partition of unity (fα) subordinate to that covering such that P α fα(y) = 1, y ∈ H (note that the sum is finite). Similarly for ϕ, thus limiting ourselves to domains of a pair of charts. By [Gösser,Kunzinger & al., Chapter 3.1.4], we can identify distributions localised on chart domains with distributions on subsets of Rd and Rr. Thus, in what follows we shall assume that X and Y are open subsets of Rd and Rr.

16 for m ﬁnite, while for m = ∞ we should modify the above to

0 ∞ β (∃ m ∈ N)(∃ C > 0)(∀ ψ ∈ CH (Y )) |hKϕ, ψi| C max k∂ ψkL∞(H) . 6 0 |β|6m

β |hKϕ, ψi| 6 C max k∂ ψkL∞(H) , |β|6m

16 From kernel to operator (a) l m ϕ ∈ Cc(X); prove the continuity of Kϕ on Cc (Y ) (it is clearly linear since the tensor product is bilinear, while K is linear). m i.e. for H ∈ K(Y ), mapping ψ 7→ hKϕ, ψi is a cont. lin. funct. on CH (Y ). We can assume X and Y to be open subsets of Rd and Rr. Indeed, first take an open covering of Y , consisting of chart domains, and a partition of unity (fα) subordinate to that covering such that P α fα(y) = 1, y ∈ H (note that the sum is finite). Similarly for ϕ, thus limiting ourselves to domains of a pair of charts. By [Gösser,Kunzinger & al., Chapter 3.1.4], we can identify distributions localised on chart domains with distributions on subsets of Rd and Rr. Thus, in what follows we shall assume that X and Y are open subsets of Rd and Rr. We shall therefore show that there exists a constant C > 0 such that for any m ψ ∈ CH (Y ) it holds

β |hKϕ, ψi| 6 C max k∂ ψkL∞(H) , |β|6m for m ﬁnite, while for m = ∞ we should modify the above to

0 ∞ β (∃ m ∈ N)(∃ C > 0)(∀ ψ ∈ CH (Y )) |hKϕ, ψi| C max k∂ ψkL∞(H) . 6 0 |β|6m

16 by taking M to be of the form L × H, with L ∈ K(X), and Ψ = ϕ ψ such that supp ϕ ⊆ L, we have ˜ α β |hKϕ, ψi| = |hK, ϕ ψi| 6 C max k∂ ϕ ∂ ψkL∞(L×H) |α|6l,|β|6m ˜ α β β 6 C max k∂ ϕkL∞(L) max k∂ ψkL∞(H) 6 C max k∂ ψkL∞(H) , |α|6l |β|6m |β|6m

0 and therefore Kϕ ∈ Dm(Y ).

From kernel to operator (a)(cont.)

17 From kernel to operator (a)(cont.)

K is a distribution of anisotropic order on X × Y : ˜ l,m (∀ M ∈ K(X × Y ))(∃C > 0)(∀Ψ ∈ Cc (X × Y )) ˜ α,β supp Ψ ⊆ M =⇒ |hK, Ψi| 6 C max k∂ ΨkL∞(M) , |α|6l,|β|6m with obvious modiﬁcations if either l or m is inﬁnite, by taking M to be of the form L × H, with L ∈ K(X), and Ψ = ϕ ψ such that supp ϕ ⊆ L, we have ˜ α β |hKϕ, ψi| = |hK, ϕ ψi| 6 C max k∂ ϕ ∂ ψkL∞(L×H) |α|6l,|β|6m ˜ α β β 6 C max k∂ ϕkL∞(L) max k∂ ψkL∞(H) 6 C max k∂ ψkL∞(H) , |α|6l |β|6m |β|6m

0 and therefore Kϕ ∈ Dm(Y ).

17 m For continuity, take an arbitrary L ∈ K(X) and an arbitrary ψ ∈ Cc (Y ). We need to show the existence of C¯ > 0 such that ¯ α |hKϕ, ψi| 6 C max k∂ ϕkL∞(L) . |α|6l However, we have already shown that above: just take ¯ ˜ β C = C max k∂ ψkL∞(H) . |β|6m

l 0 Therefore, the mapping ϕ 7→ Kϕ, from Cc(X) to Dm(Y ) is linear and continuous.

From kernel to operator (a)(cont.)

The linearity of mapping ϕ 7→ Kϕ readily follows from the bilinearity of tensor product and the linearity of K.

18 However, we have already shown that above: just take ¯ ˜ β C = C max k∂ ψkL∞(H) . |β|6m

l 0 Therefore, the mapping ϕ 7→ Kϕ, from Cc(X) to Dm(Y ) is linear and continuous.

From kernel to operator (a)(cont.)

The linearity of mapping ϕ 7→ Kϕ readily follows from the bilinearity of tensor product and the linearity of K. m For continuity, take an arbitrary L ∈ K(X) and an arbitrary ψ ∈ Cc (Y ). We need to show the existence of C¯ > 0 such that ¯ α |hKϕ, ψi| 6 C max k∂ ϕkL∞(L) . |α|6l

18 From kernel to operator (a)(cont.)

l 0 Therefore, the mapping ϕ 7→ Kϕ, from Cc(X) to Dm(Y ) is linear and continuous.

18 The proof of existence will be divided into two steps. In the first step we assume that X and Y are open subsets of Rd and Rr, and additionally, that 0 the range of A is C(Y ) ⊆ Dm(Y ) (understood as distributions which can be identified with continuous functions). This will allow us to write explicitly the m action of Aϕ on a test function ψ ∈ Cc (Y ), which will finally enable us to define the kernel K. In the second step, we shall use a partition of unity and the structure theorem of distributions to reduce the problem to the first step.

From operator to kernel (b): uniqueness and overview

Let us ﬁrst prove the uniqueness. By formula

hK, ϕψi = hKϕ, ψi = hAϕ, ψi , ∞ ∞ a continuous functional K on Cc (X) Cc (Y ) is deﬁned. As it is deﬁned on ∞ a dense subset of Cc (X × Y ), such K is uniquely determined on the whole ∞ Cc (X × Y ).

19 From operator to kernel (b): uniqueness and overview

Let us ﬁrst prove the uniqueness. By formula

hK, ϕψi = hKϕ, ψi = hAϕ, ψi , ∞ ∞ a continuous functional K on Cc (X) Cc (Y ) is defined. As it is defined on ∞ a dense subset of Cc (X × Y ), such K is uniquely determined on the whole ∞ Cc (X × Y ). The proof of existence will be divided into two steps. In the first step we assume that X and Y are open subsets of Rd and Rr, and additionally, that 0 the range of A is C(Y ) ⊆ Dm(Y ) (understood as distributions which can be identified with continuous functions). This will allow us to write explicitly the m action of Aϕ on a test function ψ ∈ Cc (Y ), which will finally enable us to define the kernel K. In the second step, we shall use a partition of unity and the structure theorem of distributions to reduce the problem to the first step.

19 m Its action on a test function ψ ∈ Cc (Y ) is given by Z hAϕ, ψi = (Aϕ)(y)ψ(y)dy . Y

l Continuity of A implies that A :Cc(X) −→ C(Y ) is continuous when the 0 range is equipped with the weak ∗ topology inherited from Dm(Y ). As the latter is a Hausdorﬀ space, that operator has a closed graph, but this remains true even when we replace the topology on C(Y ) by its standard Fr´echettopology [Narici & Beckenstein, Exercise 14.101(a)], which is stronger. Now we can apply the Closed graph theorem [Narici & Beckenstein, Theorem l 14.3.4(b)], as Cc(X) is barreled, as a strict inductive limit of barreled spaces, l to conclude that A :Cc(X) −→ C(Y ) is continuous with usual strong topologies on its domain and range.

From operator to kernel (b): existence under additional assumptions

Additionally assume that X and Y are open and bounded subsets of euclidean l spaces, and that for each ϕ ∈ Cc(X), Aϕ ∈ C(Y ).

20 As the latter is a Hausdorﬀ space, that operator has a closed graph, but this remains true even when we replace the topology on C(Y ) by its standard Fr´echettopology [Narici & Beckenstein, Exercise 14.101(a)], which is stronger. Now we can apply the Closed graph theorem [Narici & Beckenstein, Theorem l 14.3.4(b)], as Cc(X) is barreled, as a strict inductive limit of barreled spaces, l to conclude that A :Cc(X) −→ C(Y ) is continuous with usual strong topologies on its domain and range.

From operator to kernel (b): existence under additional assumptions

Additionally assume that X and Y are open and bounded subsets of euclidean l spaces, and that for each ϕ ∈ Cc(X), Aϕ ∈ C(Y ). m Its action on a test function ψ ∈ Cc (Y ) is given by Z hAϕ, ψi = (Aϕ)(y)ψ(y)dy . Y

l Continuity of A implies that A :Cc(X) −→ C(Y ) is continuous when the 0 range is equipped with the weak ∗ topology inherited from Dm(Y ).

20 Now we can apply the Closed graph theorem [Narici & Beckenstein, Theorem l 14.3.4(b)], as Cc(X) is barreled, as a strict inductive limit of barreled spaces, l to conclude that A :Cc(X) −→ C(Y ) is continuous with usual strong topologies on its domain and range.

From operator to kernel (b): existence under additional assumptions

20 From operator to kernel (b): existence under additional assumptions

20 Since Aϕ is a continuous function, Fy is well-defined and continuous as a 0 composition of continuous mappings, thus a distribution in Dl(X). l,0 Take a test function Ψ ∈ Cc (X × Y ), and fix its second variable (get a l function from Cc(X)) and apply Fy; we are interested in the properties of this mapping: y 7→ Fy(Ψ(·, y)) = AΨ(·, y) (y) . Clearly, it is well defined on Y , with a compact support contained in the projection πY (supp Ψ). Furthermore, we have: Fy(Ψ(·, y)) = AΨ(·, y) (y) kAΨ(·, y)k ∞ 6 L (πY (supp Ψ))

6 CkΨ(·, y)kCl (X) 6 CkΨkCl,0 (X×Y ) . πY (supp Ψ) supp Ψ

(b): existence under additional assumptions (cont.)

l For y ∈ Y consider a linear functional Fy :Cc(X) −→ C deﬁned by

Fy(ϕ) = (Aϕ)(y) .

21 l,0 Take a test function Ψ ∈ Cc (X × Y ), and ﬁx its second variable (get a l function from Cc(X)) and apply Fy; we are interested in the properties of this mapping: y 7→ Fy(Ψ(·, y)) = AΨ(·, y) (y) . Clearly, it is well deﬁned on Y , with a compact support contained in the projection πY (supp Ψ). Furthermore, we have: Fy(Ψ(·, y)) = AΨ(·, y) (y) kAΨ(·, y)k ∞ 6 L (πY (supp Ψ))

6 CkΨ(·, y)kCl (X) 6 CkΨkCl,0 (X×Y ) . πY (supp Ψ) supp Ψ

(b): existence under additional assumptions (cont.)

l For y ∈ Y consider a linear functional Fy :Cc(X) −→ C deﬁned by

Fy(ϕ) = (Aϕ)(y) .

Since Aϕ is a continuous function, Fy is well-deﬁned and continuous as a 0 composition of continuous mappings, thus a distribution in Dl(X).

21 Clearly, it is well deﬁned on Y , with a compact support contained in the projection πY (supp Ψ). Furthermore, we have: Fy(Ψ(·, y)) = AΨ(·, y) (y) kAΨ(·, y)k ∞ 6 L (πY (supp Ψ))

6 CkΨ(·, y)kCl (X) 6 CkΨkCl,0 (X×Y ) . πY (supp Ψ) supp Ψ

(b): existence under additional assumptions (cont.)

l For y ∈ Y consider a linear functional Fy :Cc(X) −→ C deﬁned by

Fy(ϕ) = (Aϕ)(y) .

21 (b): existence under additional assumptions (cont.)

l For y ∈ Y consider a linear functional Fy :Cc(X) −→ C deﬁned by

Fy(ϕ) = (Aϕ)(y) .

Since Aϕ is a continuous function, Fy is well-defined and continuous as a 0 composition of continuous mappings, thus a distribution in Dl(X). l,0 Take a test function Ψ ∈ Cc (X × Y ), and fix its second variable (get a l function from Cc(X)) and apply Fy; we are interested in the properties of this mapping: y 7→ Fy(Ψ(·, y)) = AΨ(·, y) (y) . Clearly, it is well defined on Y , with a compact support contained in the projection πY (supp Ψ). Furthermore, we have: Fy(Ψ(·, y)) = AΨ(·, y) (y) kAΨ(·, y)k ∞ 6 L (πY (supp Ψ))

6 CkΨ(·, y)kCl (X) 6 CkΨkCl,0 (X×Y ) . πY (supp Ψ) supp Ψ

21 α l This is also valid for ∂x Ψ, where |α| 6 l, thus Ψ(·, yn) −→ Ψ(·, y) in Cc(X). As A is continuous, the convergence is carried to C(Y ), i.e. to uniform convergence on compacts of a sequence of functions AΨ(·, yn) to AΨ(·, y). In particular,(AΨ(·, yn))(y¯) − (AΨ(·, y))(y¯) is arbitrary small independently of y¯ ∈ L, for large enough n. On the other hand, AΨ(·, y) is uniformly continuous, thus (AΨ(·, y))(y¯) − (AΨ(·, y))(y) is small for large n, independetly of y¯ ∈ L. In other terms, we have the required convergence

Fyn (Ψ(·, yn)) −→ Fy(Ψ(·, y)) . A continuous function with compact support is summable, so we can deﬁne K on Cl,0(X × Y ): c Z hK, Ψi = Fy(Ψ(·, y)) dy , Y

which is obviously linear in Ψ, as Fy is.

(b): existence under additional assumptions (cont.)

We show sequential continuity: take a sequence yn → y in Y . Denote H = πX (supp Ψ) and let L ⊆ Y be a compact such that yn, y ∈ L; Ψ is uniformly continuous on compact H × L.

22 As A is continuous, the convergence is carried to C(Y ), i.e. to uniform convergence on compacts of a sequence of functions AΨ(·, yn) to AΨ(·, y). In particular,(AΨ(·, yn))(y¯) − (AΨ(·, y))(y¯) is arbitrary small independently of y¯ ∈ L, for large enough n. On the other hand, AΨ(·, y) is uniformly continuous, thus (AΨ(·, y))(y¯) − (AΨ(·, y))(y) is small for large n, independetly of y¯ ∈ L. In other terms, we have the required convergence

Fyn (Ψ(·, yn)) −→ Fy(Ψ(·, y)) . A continuous function with compact support is summable, so we can deﬁne K on Cl,0(X × Y ): c Z hK, Ψi = Fy(Ψ(·, y)) dy , Y

which is obviously linear in Ψ, as Fy is.

(b): existence under additional assumptions (cont.)

22 On the other hand, AΨ(·, y) is uniformly continuous, thus (AΨ(·, y))(y¯) − (AΨ(·, y))(y) is small for large n, independetly of y¯ ∈ L. In other terms, we have the required convergence

Fyn (Ψ(·, yn)) −→ Fy(Ψ(·, y)) . A continuous function with compact support is summable, so we can deﬁne K on Cl,0(X × Y ): c Z hK, Ψi = Fy(Ψ(·, y)) dy , Y

which is obviously linear in Ψ, as Fy is.

(b): existence under additional assumptions (cont.)

We show sequential continuity: take a sequence yn → y in Y . Denote H = πX (supp Ψ) and let L ⊆ Y be a compact such that yn, y ∈ L; Ψ is uniformly continuous on compact H × L. α l This is also valid for ∂x Ψ, where |α| 6 l, thus Ψ(·, yn) −→ Ψ(·, y) in Cc(X). As A is continuous, the convergence is carried to C(Y ), i.e. to uniform convergence on compacts of a sequence of functions AΨ(·, yn) to AΨ(·, y). In particular,(AΨ(·, yn))(y¯) − (AΨ(·, y))(y¯) is arbitrary small independently of y¯ ∈ L, for large enough n.

22 A continuous function with compact support is summable, so we can deﬁne K on Cl,0(X × Y ): c Z hK, Ψi = Fy(Ψ(·, y)) dy , Y

which is obviously linear in Ψ, as Fy is.

(b): existence under additional assumptions (cont.)

Fyn (Ψ(·, yn)) −→ Fy(Ψ(·, y)) .

22 (b): existence under additional assumptions (cont.)

Fyn (Ψ(·, yn)) −→ Fy(Ψ(·, y)) . A continuous function with compact support is summable, so we can deﬁne K on Cl,0(X × Y ): c Z hK, Ψi = Fy(Ψ(·, y)) dy , Y

which is obviously linear in Ψ, as Fy is.

22 However, we can check that K is continuous at zero (modiﬁcations for l = ∞):

l,0 (∀ H ∈ K(X))(∀ L ∈ K(Y ))(∃ C > 0)(∀ Ψ ∈ Cc (X × Y )) supp Ψ ⊆ H × L =⇒ |hK, Ψi| 6 CkΨk l,0 . CK×L(X×Y )

l The continuity of A :Cc(X) −→ C(Y ), for Ψ supported in H × L and the fact that the support of AΨ(·, y) is contained in L gives us the estimate Z

Fy(Ψ(·, y)) dy 6 (volL)CkΨkCl,0 (X×Y ) , Y K×L as needed. ∞ ∞ Finally, it is easy to check that for ϕ ∈ Cc (X) and ψ ∈ Cc (Y ), we have: Z Z hK, ϕ ψi = Fy(ϕ ψ(y))dy = Fy(ϕ)ψ(y)dy Y Y Z = (Aϕ)(y)ψ(y)dy = hAϕ, ψi . Y

(b): existence under additional assumptions (cont.)

For continuity of K, we cannot follow [Dieudonne, 23.9.2], as our spaces are not Montel.

23 l The continuity of A :Cc(X) −→ C(Y ), for Ψ supported in H × L and the fact that the support of AΨ(·, y) is contained in L gives us the estimate Z

(b): existence under additional assumptions (cont.)

For continuity of K, we cannot follow [Dieudonne, 23.9.2], as our spaces are not Montel. However, we can check that K is continuous at zero (modiﬁcations for l = ∞):

l,0 (∀ H ∈ K(X))(∀ L ∈ K(Y ))(∃ C > 0)(∀ Ψ ∈ Cc (X × Y )) supp Ψ ⊆ H × L =⇒ |hK, Ψi| 6 CkΨk l,0 . CK×L(X×Y )

23 ∞ ∞ Finally, it is easy to check that for ϕ ∈ Cc (X) and ψ ∈ Cc (Y ), we have: Z Z hK, ϕ ψi = Fy(ϕ ψ(y))dy = Fy(ϕ)ψ(y)dy Y Y Z = (Aϕ)(y)ψ(y)dy = hAϕ, ψi . Y

(b): existence under additional assumptions (cont.)

For continuity of K, we cannot follow [Dieudonne, 23.9.2], as our spaces are not Montel. However, we can check that K is continuous at zero (modiﬁcations for l = ∞):

l,0 (∀ H ∈ K(X))(∀ L ∈ K(Y ))(∃ C > 0)(∀ Ψ ∈ Cc (X × Y )) supp Ψ ⊆ H × L =⇒ |hK, Ψi| 6 CkΨk l,0 . CK×L(X×Y )

l The continuity of A :Cc(X) −→ C(Y ), for Ψ supported in H × L and the fact that the support of AΨ(·, y) is contained in L gives us the estimate Z

Fy(Ψ(·, y)) dy 6 (volL)CkΨkCl,0 (X×Y ) , Y K×L as needed.

23 (b): existence under additional assumptions (cont.)

For continuity of K, we cannot follow [Dieudonne, 23.9.2], as our spaces are not Montel. However, we can check that K is continuous at zero (modiﬁcations for l = ∞):

l,0 (∀ H ∈ K(X))(∀ L ∈ K(Y ))(∃ C > 0)(∀ Ψ ∈ Cc (X × Y )) supp Ψ ⊆ H × L =⇒ |hK, Ψi| 6 CkΨk l,0 . CK×L(X×Y )

l The continuity of A :Cc(X) −→ C(Y ), for Ψ supported in H × L and the fact that the support of AΨ(·, y) is contained in L gives us the estimate Z

23 It is suﬃcient to show existence of distributions Kαβ on Uα × Vβ , which satisfy

∞ ∞ hAϕ, ψi = hKαβ , ϕ ψi , ϕ ∈ Cc (Uα), ψ ∈ Cc (Vβ ) . Indeed, the uniqueness of K ∈ D0(X × Y ) then follows from the fact that two distributions Kαβ and Kγδ will coincide on open sets (Uα ∩ Uγ) × (Vβ ∩ Vδ) of X × Y , while the existence of K will be a result of the localisation theorem [Dieudonne, 17.4.2].

Furthermore, if we assume that Uα and Vβ lie within domains of some charts of X and Y , in the light of results of [G¨osser,Kunzinger & al., Chapter 3.1.4], we can identify the distributions localised to these chart domains with distributions on open subsets of Rd. Thus, without loss of generality, we assume that U and V are relatively compact open subsets of Rd.

(b) existence in general: reduction to charts

Let (Uα) and (Vβ ) be covers consisting of relatively compact open sets.

24 Indeed, the uniqueness of K ∈ D0(X × Y ) then follows from the fact that two distributions Kαβ and Kγδ will coincide on open sets (Uα ∩ Uγ) × (Vβ ∩ Vδ) of X × Y , while the existence of K will be a result of the localisation theorem [Dieudonne, 17.4.2].

(b) existence in general: reduction to charts

Let (Uα) and (Vβ ) be covers consisting of relatively compact open sets.

It is suﬃcient to show existence of distributions Kαβ on Uα × Vβ , which satisfy

∞ ∞ hAϕ, ψi = hKαβ , ϕ ψi , ϕ ∈ Cc (Uα), ψ ∈ Cc (Vβ ) .

24 Furthermore, if we assume that Uα and Vβ lie within domains of some charts of X and Y , in the light of results of [G¨osser,Kunzinger & al., Chapter 3.1.4], we can identify the distributions localised to these chart domains with distributions on open subsets of Rd. Thus, without loss of generality, we assume that U and V are relatively compact open subsets of Rd.

(b) existence in general: reduction to charts

Let (Uα) and (Vβ ) be covers consisting of relatively compact open sets.

It is suﬃcient to show existence of distributions Kαβ on Uα × Vβ , which satisfy

24 (b) existence in general: reduction to charts

Let (Uα) and (Vβ ) be covers consisting of relatively compact open sets.

It is suﬃcient to show existence of distributions Kαβ on Uα × Vβ , which satisfy

24 A˜ is well-deﬁned, and by the assumptions continuous. Take a relatively compact open neighbourhood W of Cl V in Y and pick a smooth cut-oﬀ function ρ being one on Cl V and supported in W . l ˜ 0 For ϕ ∈ Cc(U), ρAϕ ∈ Dm(W ) and has a compact support. Next we use the Structure theorem for distributions: from its proof [Friedlander & Joshi, Theorem 5.4.1], we can write

˜ m+2 m+2 ˜ ρAϕ = ∂1 . . . ∂d Em+2 ∗ (ρAϕ) ,

m+2 m+2 where Em+2 is the fundamental solution of ∂1 . . . ∂d (derivatives in y), m+2 m+2 i.e. it satisﬁes the equation ∂1 . . . ∂d Em+2 = δ0 (explicit formula for Em+2 in loc.cit.), and Em+2 ∗ (ρAϕ˜ ) is a continuous function. l m Denoting by Eem+2∗ the transpose of Em+2∗, for ϕ ∈ Cc(U) and ψ ∈ Cc (W ) D ˜ E D ˜ E Em+2 ∗ (ρAϕ), ψ = Aϕ, ρEem+2 ∗ ψ ,

˜ l 0 concluding that ϕ 7→ Em+2 ∗ (ρAϕ) is continuous from Cc(U) to Dm(W ).

(b) existence in general: the structure theorem ˜ l 0 l m Consider A :Cc(U) → Dm(V ) deﬁned by: for ϕ ∈ Cc(U) and ψ ∈ Cc (V ) hAϕ,˜ ψi = hAϕ, ψi .

25 Take a relatively compact open neighbourhood W of Cl V in Y and pick a smooth cut-oﬀ function ρ being one on Cl V and supported in W . l ˜ 0 For ϕ ∈ Cc(U), ρAϕ ∈ Dm(W ) and has a compact support. Next we use the Structure theorem for distributions: from its proof [Friedlander & Joshi, Theorem 5.4.1], we can write

˜ m+2 m+2 ˜ ρAϕ = ∂1 . . . ∂d Em+2 ∗ (ρAϕ) ,

˜ l 0 concluding that ϕ 7→ Em+2 ∗ (ρAϕ) is continuous from Cc(U) to Dm(W ).

(b) existence in general: the structure theorem ˜ l 0 l m Consider A :Cc(U) → Dm(V ) deﬁned by: for ϕ ∈ Cc(U) and ψ ∈ Cc (V ) hAϕ,˜ ψi = hAϕ, ψi .

A˜ is well-deﬁned, and by the assumptions continuous.

25 l ˜ 0 For ϕ ∈ Cc(U), ρAϕ ∈ Dm(W ) and has a compact support. Next we use the Structure theorem for distributions: from its proof [Friedlander & Joshi, Theorem 5.4.1], we can write

˜ m+2 m+2 ˜ ρAϕ = ∂1 . . . ∂d Em+2 ∗ (ρAϕ) ,

˜ l 0 concluding that ϕ 7→ Em+2 ∗ (ρAϕ) is continuous from Cc(U) to Dm(W ).

(b) existence in general: the structure theorem ˜ l 0 l m Consider A :Cc(U) → Dm(V ) deﬁned by: for ϕ ∈ Cc(U) and ψ ∈ Cc (V ) hAϕ,˜ ψi = hAϕ, ψi .

A˜ is well-deﬁned, and by the assumptions continuous. Take a relatively compact open neighbourhood W of Cl V in Y and pick a smooth cut-oﬀ function ρ being one on Cl V and supported in W .

25 l m Denoting by Eem+2∗ the transpose of Em+2∗, for ϕ ∈ Cc(U) and ψ ∈ Cc (W ) D ˜ E D ˜ E Em+2 ∗ (ρAϕ), ψ = Aϕ, ρEem+2 ∗ ψ ,

˜ l 0 concluding that ϕ 7→ Em+2 ∗ (ρAϕ) is continuous from Cc(U) to Dm(W ).

(b) existence in general: the structure theorem ˜ l 0 l m Consider A :Cc(U) → Dm(V ) deﬁned by: for ϕ ∈ Cc(U) and ψ ∈ Cc (V ) hAϕ,˜ ψi = hAϕ, ψi .

A˜ is well-deﬁned, and by the assumptions continuous. Take a relatively compact open neighbourhood W of Cl V in Y and pick a smooth cut-oﬀ function ρ being one on Cl V and supported in W . l ˜ 0 For ϕ ∈ Cc(U), ρAϕ ∈ Dm(W ) and has a compact support. Next we use the Structure theorem for distributions: from its proof [Friedlander & Joshi, Theorem 5.4.1], we can write

˜ m+2 m+2 ˜ ρAϕ = ∂1 . . . ∂d Em+2 ∗ (ρAϕ) ,

25 (b) existence in general: the structure theorem ˜ l 0 l m Consider A :Cc(U) → Dm(V ) deﬁned by: for ϕ ∈ Cc(U) and ψ ∈ Cc (V ) hAϕ,˜ ψi = hAϕ, ψi .

˜ m+2 m+2 ˜ ρAϕ = ∂1 . . . ∂d Em+2 ∗ (ρAϕ) ,

˜ l 0 concluding that ϕ 7→ Em+2 ∗ (ρAϕ) is continuous from Cc(U) to Dm(W ). 25 ∞ ∞ Taking ϕ ∈ Cc (U) and ψ ∈ Cc (V ), we have

m+2 m+2 D ˜ m+2 m+2 E R, ϕ ∂1 . . . ∂d ψ = Em+2 ∗ (ρAϕ), ∂1 . . . ∂d ψ d(m+2) D m+2 m+2 ˜ E = (−1) ∂1 . . . ∂d Em+2 ∗ (ρAϕ) , ψ D E = (−1)d(m+2) ρAϕ,˜ ψ D E = (−1)d(m+2) Aϕ,˜ ρψ

= (−1)d(m+2)hAϕ, ψi,

d(m+2) m+2 m+2 which gives hAϕ, ψi = (−1) ∂1 . . . ∂d R, ϕ ψ , where the derivatives are taken with respect to the variable y. Since R was an element of 0 0 Dl,0(U × W ), we conclude that A ∈ Dl,d(m+2)(U × V ).

(b) existence in general: reduction to special case

0 ∞ Now we can ﬁnd R ∈ Dl,0(U × W ) such that for all ϕ ∈ Cc (U) and ∞ ψ ∈ Cc (W ) it holds

D ˜ E Em+2 ∗ (ρAϕ), ψ = hR, ϕ ψi .

26 (b) existence in general: reduction to special case

0 ∞ Now we can ﬁnd R ∈ Dl,0(U × W ) such that for all ϕ ∈ Cc (U) and ∞ ψ ∈ Cc (W ) it holds

D ˜ E Em+2 ∗ (ρAϕ), ψ = hR, ϕ ψi .

∞ ∞ Taking ϕ ∈ Cc (U) and ψ ∈ Cc (V ), we have

= (−1)d(m+2)hAϕ, ψi,

26 If one used a more constructive proof of the Schwartz kernel theorem, for example [Simanca, Theorem 1.3.4], one would end up increasing the order with respect to both variables x and y. This occurs naturally, because one needs to secure the integrability of the function which is used to deﬁne the kernel function. One interesting approach to the kernel theorem is given in [Tr`eves,Chapter 51]. This approach is based on deep results of functional analysis on tensor products of nuclear spaces of Alexander Grothendieck. This approach might result in further improvements of the preceeding theorem. This is a subject of our current ongoing research.

Remarks

27 One interesting approach to the kernel theorem is given in [Tr`eves,Chapter 51]. This approach is based on deep results of functional analysis on tensor products of nuclear spaces of Alexander Grothendieck. This approach might result in further improvements of the preceeding theorem. This is a subject of our current ongoing research.

Remarks

0 Note that in part (b) we did not get K ∈ Dl,m(X × Y ), as one would expect. The order with respect to x variable remained the same, but the order with respect to y increased from m to d(m + 2). Interchanging the roles of X and 0 Y , the same proof gives K ∈ Dd(l+2),m(X × Y ), where order with respect to y remained the same, but order with respect to the x variable increased from l to d(l + 2). Since uniqueness of K ∈ D0(X × Y ) has already been determined, we 0 0 conclude that K ∈ Dl,d(m+2)(X × Y ) ∩ Dd(l+2),m(X × Y ). It might be interesting to see some additional properties of that intersection. If one used a more constructive proof of the Schwartz kernel theorem, for example [Simanca, Theorem 1.3.4], one would end up increasing the order with respect to both variables x and y. This occurs naturally, because one needs to secure the integrability of the function which is used to deﬁne the kernel function.

27 Remarks

27 Indeed, we already have µ ∈ D0(Rd × Sd−1) and the following bound with ϕ := ϕ1ϕ2:

|hµ, ϕ ψi| CkψkCκ(Sd−1)kϕkC (Rd) , 6 Kl where C does not depend on ϕ and ψ. Now we just need to apply the Schwartz kernel theorem given above to 0,d(κ+2) d d−1 conclude that µ is a continuous linear functional on Cc (R × S ).

Consequence for H-distributions

28 Now we just need to apply the Schwartz kernel theorem given above to 0,d(κ+2) d d−1 conclude that µ is a continuous linear functional on Cc (R × S ).

Consequence for H-distributions

By the previous theorem the H-distribution µ mentioned at the beginning 0 d d−1 belongs to the space D0,d(κ+2)(R × S ), i.e. it is a distribution of order 0 in x and of order not more than d(κ + 2) in ξ. Indeed, we already have µ ∈ D0(Rd × Sd−1) and the following bound with ϕ := ϕ1ϕ2:

|hµ, ϕ ψi| CkψkCκ(Sd−1)kϕkC (Rd) , 6 Kl where C does not depend on ϕ and ψ.

28 Consequence for H-distributions

28 Thank you for your attention.