Contributions to Analysis and Functional Analysis in memoriam Pawe - - PowerPoint PPT Presentation

Contributions to Analysis and Functional Analysis

in memoriam

Pawe l Doma´ nski

Dietmar Vogt

Bergische Universit¨ at Wuppertal Pawe l Doma´ nski Memorial Conference Bedlewo 2018

Contributions to:

◮ Real analytic functions

• Linear structure, subspaces, quotient spaces, bases
• Algebra structure: ideals, algebra/composition operators
• Vector valued real analytic functions

◮ Homological theory, Ext, Proj

• Interplay with vector valued real analytic functions, solutions

with parameter, interpolation

• Special invariants, exactness of tensorized sequences
• Splitting of differential complexes

◮ Special operators

• Structural theory of Hadamard-/Euler-operators on real

analytic functions

• Solvability of Hadamard-/Euler-operators on real analytic or

smooth functions

Contributions to:

◮ Real analytic functions

• Linear structure, subspaces, quotient spaces, bases
• Algebra structure: ideals, algebra/composition operators
• Vector valued real analytic functions

◮ Homological theory, Ext, Proj

• Interplay with vector valued real analytic functions, solutions

with parameter, interpolation

• Special invariants, exactness of tensorized sequences
• Splitting of differential complexes

◮ Special operators

• Structural theory of Hadamard-/Euler-operators on real

analytic functions

• Solvability of Hadamard-/Euler-operators on real analytic or

smooth functions Many more results on:

Theory of functional analysis, dynamics of operators, vector-valued hyper-functions, abstract Cauchy-problem, ...

Real analytic functions

Let Ω ⊂ Rd be open. A real analytic function is an infinitely differentiable function which can be expanded around every point in Ω into its Taylor series or, equivalently, which can be extended to a holomorphic function on some complex neighborhood of Ω.

Real analytic functions

Let Ω ⊂ Rd be open. A real analytic function is an infinitely differentiable function which can be expanded around every point in Ω into its Taylor series or, equivalently, which can be extended to a holomorphic function on some complex neighborhood of Ω. A(Ω) denotes the space of real analytic functions on Ω. It is an algebra over C. It carries a unique natural topology which makes it a topological algebra. This is given by A(Ω) = projnH(Kn) = indωH(ω) where K1 ⊂ K2 ⊂ . . . is a compact exhaustion of Ω and ω runs through the complex neighborhoods of Ω.

Real analytic functions

Let Ω ⊂ Rd be open. A real analytic function is an infinitely differentiable function which can be expanded around every point in Ω into its Taylor series or, equivalently, which can be extended to a holomorphic function on some complex neighborhood of Ω. A(Ω) denotes the space of real analytic functions on Ω. It is an algebra over C. It carries a unique natural topology which makes it a topological algebra. This is given by A(Ω) = projnH(Kn) = indωH(ω) where K1 ⊂ K2 ⊂ . . . is a compact exhaustion of Ω and ω runs through the complex neighborhoods of Ω. With this topology A(Ω) is complete, nuclear, ultrabornological (PDF)-space.

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms.

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms. ◮ E subspace of A(Ω) ⇒ E ∈ (DN). ◮ E quotient space of A(Ω) ⇒ E ∈ (Ω).

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms. ◮ E subspace of A(Ω) ⇒ E ∈ (DN). ◮ E quotient space of A(Ω) ⇒ E ∈ (Ω). Examples?

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms. ◮ E subspace of A(Ω) ⇒ E ∈ (DN). ◮ E quotient space of A(Ω) ⇒ E ∈ (Ω). Examples? Fr´ echet subspaces: zero solutions of an elliptic operator

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms. ◮ E subspace of A(Ω) ⇒ E ∈ (DN). ◮ E quotient space of A(Ω) ⇒ E ∈ (Ω). Examples? Fr´ echet subspaces: zero solutions of an elliptic operator Fr´ echet quotient spaces: they exist (difficult to show)

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms. ◮ E subspace of A(Ω) ⇒ E ∈ (DN). ◮ E quotient space of A(Ω) ⇒ E ∈ (Ω). Examples? Fr´ echet subspaces: zero solutions of an elliptic operator Fr´ echet quotient spaces: they exist (difficult to show) ◮ Precise characterization of Fr´ echet subspaces and quotients by invariants and nuclearity-type

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms. ◮ E subspace of A(Ω) ⇒ E ∈ (DN). ◮ E quotient space of A(Ω) ⇒ E ∈ (Ω). Examples? Fr´ echet subspaces: zero solutions of an elliptic operator Fr´ echet quotient spaces: they exist (difficult to show) ◮ Precise characterization of Fr´ echet subspaces and quotients by invariants and nuclearity-type Theorem: E complemented Fr´ echet subspace of A(Ω) ⇒ E finite dimensional.

Fr´ echet sub- and quotient spaces of A(Ω)

E Fr´ echet space, · 1 ≤ · 2 ≤ . . . semi-norms. ◮ E subspace of A(Ω) ⇒ E ∈ (DN). ◮ E quotient space of A(Ω) ⇒ E ∈ (Ω). Examples? Fr´ echet subspaces: zero solutions of an elliptic operator Fr´ echet quotient spaces: they exist (difficult to show) ◮ Precise characterization of Fr´ echet subspaces and quotients by invariants and nuclearity-type Theorem: E complemented Fr´ echet subspace of A(Ω) ⇒ E finite dimensional. Consequence: Improvement of the Grothendieck-Poly result.

Bases in A(Ω)

A basis is a sequence f1, f2, f3, . . . in A(Ω) so that every f ∈ A(Ω) has a unique expansion f = ∞

n=1 λn fn.

Bases in A(Ω)

A basis is a sequence f1, f2, f3, . . . in A(Ω) so that every f ∈ A(Ω) has a unique expansion f = ∞

n=1 λn fn.

A(Ω) is a complete separable locally convex space. Long term problem: does it have a basis, and if so, what is it?

Bases in A(Ω)

A basis is a sequence f1, f2, f3, . . . in A(Ω) so that every f ∈ A(Ω) has a unique expansion f = ∞

n=1 λn fn.

A(Ω) is a complete separable locally convex space. Long term problem: does it have a basis, and if so, what is it? Theorem: A(Ω) has no basis.

Bases in A(Ω)

A basis is a sequence f1, f2, f3, . . . in A(Ω) so that every f ∈ A(Ω) has a unique expansion f = ∞

n=1 λn fn.

A(Ω) is a complete separable locally convex space. Long term problem: does it have a basis, and if so, what is it? Theorem: A(Ω) has no basis. Argument: ◮ If A(Ω) has a basis then it is a (PLS)-K¨

• the space.

◮ An ultrabornological (PLS)-K¨

• the without infinite

dimensional complemented Fr´ echet subspaces is a (DF)-space. ◮ A(Ω) has no infinite dimensional complemented Fr´ echet subspaces but it is not a (DF)-space. ◮ A(Ω) has no basis.

Ideals in the algebra A(Rd)

J ⊂ A(Rd) closed ideal.

Ideals in the algebra A(Rd)

J ⊂ A(Rd) closed ideal. For an analytic set V in neighborhood of Rd define: JV = {f ∈ A(Rd) : fa = 0 on Va for all a ∈ X = V ∩ Rd}.

Ideals in the algebra A(Rd)

J ⊂ A(Rd) closed ideal. For an analytic set V in neighborhood of Rd define: JV = {f ∈ A(Rd) : fa = 0 on Va for all a ∈ X = V ∩ Rd}. Results: ◮ There is V such that J = JV iff Ja = Rad(Ja) for all a ∈ X.

Ideals in the algebra A(Rd)

J ⊂ A(Rd) closed ideal. For an analytic set V in neighborhood of Rd define: JV = {f ∈ A(Rd) : fa = 0 on Va for all a ∈ X = V ∩ Rd}. Results: ◮ There is V such that J = JV iff Ja = Rad(Ja) for all a ∈ X. ◮ If JV is complemented in A(Rd) then Va satisfies PLloc for all a ∈ X. ◮ If X is compact or homogeneous then the converse is true.

Ideals in the algebra A(Rd)

J ⊂ A(Rd) closed ideal. For an analytic set V in neighborhood of Rd define: JV = {f ∈ A(Rd) : fa = 0 on Va for all a ∈ X = V ∩ Rd}. Results: ◮ There is V such that J = JV iff Ja = Rad(Ja) for all a ∈ X. ◮ If JV is complemented in A(Rd) then Va satisfies PLloc for all a ∈ X. ◮ If X is compact or homogeneous then the converse is true. Va satisfies PLloc if there is a constant A > 0 such that for all holomorphic functions f on Va |f (z)| ≤ f A|Im z|

Va

f 1−A|Im z|

Xa

in a complex neighborhood of a independent of f .

Composition operators

M, N real analytic manifolds, algebra hom. A(N) → A(M):

Composition operators

M, N real analytic manifolds, algebra hom. A(N) → A(M): ϕ : M → N real analytic map, Cϕ(f ) := f ◦ ϕ.

Composition operators

M, N real analytic manifolds, algebra hom. A(N) → A(M): ϕ : M → N real analytic map, Cϕ(f ) := f ◦ ϕ. J := ker Cϕ. X = LocJ ◮ Cϕ has closed range ⇒ ϕ(M) has the global extension property (and ϕ(M) = X). ◮ Converse is true if ϕ semi-proper.

Composition operators

M, N real analytic manifolds, algebra hom. A(N) → A(M): ϕ : M → N real analytic map, Cϕ(f ) := f ◦ ϕ. J := ker Cϕ. X = LocJ ◮ Cϕ has closed range ⇒ ϕ(M) has the global extension property (and ϕ(M) = X). ◮ Converse is true if ϕ semi-proper. ◮ Cϕ is open onto its range ⇔ ϕ is semi-proper and ϕ(M) has the semi-global extension property.

Composition operators

M, N real analytic manifolds, algebra hom. A(N) → A(M): ϕ : M → N real analytic map, Cϕ(f ) := f ◦ ϕ. J := ker Cϕ. X = LocJ ◮ Cϕ has closed range ⇒ ϕ(M) has the global extension property (and ϕ(M) = X). ◮ Converse is true if ϕ semi-proper. ◮ Cϕ is open onto its range ⇔ ϕ is semi-proper and ϕ(M) has the semi-global extension property. Special case Cϕ : A(R) → A(R). ◮ Complete description of the eigenvalues and the eigenvectors

• f Cϕ.

Composition operators

M, N real analytic manifolds, algebra hom. A(N) → A(M): ϕ : M → N real analytic map, Cϕ(f ) := f ◦ ϕ. J := ker Cϕ. X = LocJ ◮ Cϕ has closed range ⇒ ϕ(M) has the global extension property (and ϕ(M) = X). ◮ Converse is true if ϕ semi-proper. ◮ Cϕ is open onto its range ⇔ ϕ is semi-proper and ϕ(M) has the semi-global extension property. Special case Cϕ : A(R) → A(R). ◮ Complete description of the eigenvalues and the eigenvectors

• f Cϕ.

◮ The Abel equation Cϕf = f + 1 has a real analytic solution iff ϕ has no fixed points and the set of critical points is bounded from above/below.

Homological concepts (local/global problem)

X = lim projjXj, Y = lim projjYj, Z = lim projjZj,

Homological concepts (local/global problem)

X = lim projjXj, Y = lim projjYj, Z = lim projjZj, exact sequence 0 → X → Y

q

→ Z

Homological concepts (local/global problem)

X = lim projjXj, Y = lim projjYj, Z = lim projjZj, exact sequence 0 → X → Y

q

→ Z Assume: induces for all j exact sequences 0 → Xj → Yj

qj

→ Zj → 0

Homological concepts (local/global problem)

X = lim projjXj, Y = lim projjYj, Z = lim projjZj, exact sequence 0 → X → Y

q

→ Z Assume: induces for all j exact sequences 0 → Xj → Yj

qj

→ Zj → 0 Problem: q surjective?

Homological concepts (local/global problem)

X = lim projjXj, Y = lim projjYj, Z = lim projjZj, exact sequence 0 → X → Y

q

→ Z Assume: induces for all j exact sequences 0 → Xj → Yj

qj

→ Zj → 0 Problem: q surjective? Yields “long exact sequence” 0 → X → Y

q

→ Z → Proj1X → Proj1Y → Proj1Z → 0.

Homological concepts (local/global problem)

X = lim projjXj, Y = lim projjYj, Z = lim projjZj, exact sequence 0 → X → Y

q

→ Z Assume: induces for all j exact sequences 0 → Xj → Yj

qj

→ Zj → 0 Problem: q surjective? Yields “long exact sequence” 0 → X → Y

q

→ Z → Proj1X → Proj1Y → Proj1Z → 0. Example: Ω ⊂ Rd open, P(∂) differential operator, K1 ⊂ K2 ⊂ compact exhaustion, P(∂) : E (Kj) → E (Kj) surjective for all j, C∞(Ω) = lim projjE (Kj), P(∂) : C∞(Ω) → C∞(Ω) surjective?

Lifting, splitting, tensors

All of the following are based on Proj1-concept.

Lifting, splitting, tensors

All of the following are based on Proj1-concept.

X Y

q

Z

E

˜ ϕ

• ϕ

Lifting, splitting, tensors

All of the following are based on Proj1-concept.

X Y

q

Z

E

˜ ϕ

• ϕ
• Assume that for all j there is ˜

ϕj with

Xj Yj

qj

Zj

E

˜ ϕj

• ϕj

Lifting, splitting, tensors

All of the following are based on Proj1-concept.

X Y

q

Z

E

˜ ϕ

• ϕ
• Assume that for all j there is ˜

ϕj with

Xj Yj

qj

Zj

E

˜ ϕj

• ϕj
• Yields for every j exact sequence

0 − → L(E, Xj) − → L(E, Yj) − → L(E, Zj) − → 0

and therefore a long exact sequence 0 → L(E, X) → L(E, Y ) → L(E, Z) → Ext1(E, X) → Ext1(E, Y ) → ..

and therefore a long exact sequence 0 → L(E, X) → L(E, Y ) → L(E, Z) → Ext1(E, X) → Ext1(E, Y ) → .. Ext1(E, X) = 0 implies under suitable assumptions:

• 1. Every ϕ ∈ L(E, Z) can be lifted under q to ˜

ϕ ∈ L(E, Y )

• 2. q ⊗ id : Y ˆ

⊗πE ′ → Z ˆ ⊗πE ′ is surjective

• 3. Every exact sequence 0 → X → Y → E → 0 splits.

and therefore a long exact sequence 0 → L(E, X) → L(E, Y ) → L(E, Z) → Ext1(E, X) → Ext1(E, Y ) → .. Ext1(E, X) = 0 implies under suitable assumptions:

• 1. Every ϕ ∈ L(E, Z) can be lifted under q to ˜

ϕ ∈ L(E, Y )

• 2. q ⊗ id : Y ˆ

⊗πE ′ → Z ˆ ⊗πE ′ is surjective

• 3. Every exact sequence 0 → X → Y → E → 0 splits.

Conditions for Ext1(E, X) = 0, X (PLS)-space: ◮ For E Fr´ echet - or (DF) space, in terms of splitting conditions (H), resp. (G) ◮ Same, in terms of invariants (P∗) for X and suitable ∗ for E, where ∗ stands for one of the submultiplicative invariants of Fr´ echet space theory.

and therefore a long exact sequence 0 → L(E, X) → L(E, Y ) → L(E, Z) → Ext1(E, X) → Ext1(E, Y ) → .. Ext1(E, X) = 0 implies under suitable assumptions:

• 1. Every ϕ ∈ L(E, Z) can be lifted under q to ˜

ϕ ∈ L(E, Y )

• 2. q ⊗ id : Y ˆ

⊗πE ′ → Z ˆ ⊗πE ′ is surjective

• 3. Every exact sequence 0 → X → Y → E → 0 splits.

Conditions for Ext1(E, X) = 0, X (PLS)-space: ◮ For E Fr´ echet - or (DF) space, in terms of splitting conditions (H), resp. (G) ◮ Same, in terms of invariants (P∗) for X and suitable ∗ for E, where ∗ stands for one of the submultiplicative invariants of Fr´ echet space theory. Condition for Proj1(X ˆ ⊗πE) = 0, X and E (PLN)-spaces, in terms of condition (T).

Splitting condition, Invariants

X = lim projNXN; XN = lim indXN,n, with norms · N,n, E analogous.

Splitting condition, Invariants

X = lim projNXN; XN = lim indXN,n, with norms · N,n, E analogous.

Condition (T):

∀ N ∃ M ∀ K ∃ n ∀ m ∃ k, S ∀ x ∈ X ′

n, y ∈ E ′ N :

x ◦ iM

n ∗ M,my ◦ iM n ∗ M,m

≤ S

• x∗

N,ny∗ N,n + x ◦ iK N ∗ K,ky ◦ iK N ∗ K,k

• .

Dual interpolation property for small θ:

∀ N ∃ M ∀ K ∃ n ∀ m ∃ θ0 ∈]0, 1[ ∀ θ ∈]0, θ0[ ∃ k, C, ∀ x′ ∈ X ′

N :

x′ ◦ iM

n ∗ M,m ≤ C x′ ◦ iK N ∗(1−θ) K,k

x′∗θ

N,n.

Property (PΩ):

∀ N ∃ M ≥ N ∀ K ≥ M ∃ n ∀ m ∃ k, C, θ ∈]0, 1[ ∀ x′ ∈ X ′

N :

x′ ◦ iM

n ∗ M,m ≤ C x′ ◦ iK N ∗(1−θ) K,k

max

• x′∗θ

N,n, x′ ◦ iK N ∗θ K,k

• .

Vector valued real analytic functions

Let E be a complete locally convex space. An E-valued function f

• n Rd is called real analytic if y ◦ f is real analytic for every y ∈ E ′.

Notation: A(Rd, E) .

Vector valued real analytic functions

Let E be a complete locally convex space. An E-valued function f

• n Rd is called real analytic if y ◦ f is real analytic for every y ∈ E ′.

Notation: A(Rd, E) . Then A(Rd, E) = E ˆ ⊗πA(Rd) .

Vector valued real analytic functions

Let E be a complete locally convex space. An E-valued function f

• n Rd is called real analytic if y ◦ f is real analytic for every y ∈ E ′.

Notation: A(Rd, E) . Then A(Rd, E) = E ˆ ⊗πA(Rd) . Problem: Can every f ∈ A(Rd, E) be expanded in Taylor series around all x? Answer: If E ∈ (DF): Yes; if E Fr´ echet : ⇔ E ∈(DN).

Vector valued real analytic functions

Let E be a complete locally convex space. An E-valued function f

• n Rd is called real analytic if y ◦ f is real analytic for every y ∈ E ′.

Notation: A(Rd, E) . Then A(Rd, E) = E ˆ ⊗πA(Rd) . Problem: Can every f ∈ A(Rd, E) be expanded in Taylor series around all x? Answer: If E ∈ (DF): Yes; if E Fr´ echet : ⇔ E ∈(DN).

Problem of interpolation:

E fixed, surjective restriction map ρ : A(Rd) → A(X). Is ρ : A(Rd, E) → A(X, E) surjective?

• E. g. X ∈ Rd discrete, ker ρ ∼

= A(Rd) .

Vector valued real analytic functions

Let E be a complete locally convex space. An E-valued function f

• n Rd is called real analytic if y ◦ f is real analytic for every y ∈ E ′.

Notation: A(Rd, E) . Then A(Rd, E) = E ˆ ⊗πA(Rd) . Problem: Can every f ∈ A(Rd, E) be expanded in Taylor series around all x? Answer: If E ∈ (DF): Yes; if E Fr´ echet : ⇔ E ∈(DN).

Problem of interpolation:

E fixed, surjective restriction map ρ : A(Rd) → A(X). Is ρ : A(Rd, E) → A(X, E) surjective?

• E. g. X ∈ Rd discrete, ker ρ ∼

= A(Rd) . Answer: If E Fr´ echet space, then surjective; If E (DF)-space, then surjective iff E ∈ (A).

Solvability with parameters

P(D) : D ′(Ω) → D ′(Ω) surjective. U real analytic manifold. f ∈ A(U, D ′(Ω)), i.e.for λ ∈ U fλ ∈ D ′(Ω), λ “parameter” Problem: Does there exist for every f a function g ∈ A(U, D ′(Ω)) such that P(D)gλ = fλ for all λ.

Solvability with parameters

P(D) : D ′(Ω) → D ′(Ω) surjective. U real analytic manifold. f ∈ A(U, D ′(Ω)), i.e.for λ ∈ U fλ ∈ D ′(Ω), λ “parameter” Problem: Does there exist for every f a function g ∈ A(U, D ′(Ω)) such that P(D)gλ = fλ for all λ. Theorem: Yes iff ker P(D) has dual interpolation estimate for small θ (U non-compact connected) or has property (PΩ) (U compact).

Solvability with parameters

P(D) : D ′(Ω) → D ′(Ω) surjective. U real analytic manifold. f ∈ A(U, D ′(Ω)), i.e.for λ ∈ U fλ ∈ D ′(Ω), λ “parameter” Problem: Does there exist for every f a function g ∈ A(U, D ′(Ω)) such that P(D)gλ = fλ for all λ. Theorem: Yes iff ker P(D) has dual interpolation estimate for small θ (U non-compact connected) or has property (PΩ) (U compact). Translation of invariants into properties of polynomial P via PL-conditions.

Solvability with parameters

P(D) : D ′(Ω) → D ′(Ω) surjective. U real analytic manifold. f ∈ A(U, D ′(Ω)), i.e.for λ ∈ U fλ ∈ D ′(Ω), λ “parameter” Problem: Does there exist for every f a function g ∈ A(U, D ′(Ω)) such that P(D)gλ = fλ for all λ. Theorem: Yes iff ker P(D) has dual interpolation estimate for small θ (U non-compact connected) or has property (PΩ) (U compact). Translation of invariants into properties of polynomial P via PL-conditions. Theorem: P homogeneous , Ω open convex. Solvability with real analytic parameters iff P(D) has a continuous linear right inverse in D ′(Ω) (equivalently: in C∞(Ω)).

Splitting of differential complexes

Fr´ echet spaces can be graded, i. e. written as projective limit in different ways: · n fundamental system of semi-norms in E.

• 1. En ‘Banach grading’,
• 2. E/ · n with quotient topology ‘strict norm grading’.

Splitting of differential complexes

Fr´ echet spaces can be graded, i. e. written as projective limit in different ways: · n fundamental system of semi-norms in E.

• 1. En ‘Banach grading’,
• 2. E/ · n with quotient topology ‘strict norm grading’.

Ω ⊂ Rd open, K1 ⊂ K2 ⊂ .. compact exhaustion, E (Kn) is strict norm grading, Proj1/Ext1-method yields:

Splitting of differential complexes

Fr´ echet spaces can be graded, i. e. written as projective limit in different ways: · n fundamental system of semi-norms in E.

• 1. En ‘Banach grading’,
• 2. E/ · n with quotient topology ‘strict norm grading’.

Ω ⊂ Rd open, K1 ⊂ K2 ⊂ .. compact exhaustion, E (Kn) is strict norm grading, Proj1/Ext1-method yields: Theorem: If Ωi ⊂ Rd are open subsets and T0 : C∞(Ω0)s → C∞(Ω1)s1 is a matrix of convolution operators such that the corresponding complex 0 → ker T0 → C∞(Ω0)s T0 → C∞(Ω1)s1 T1 → C∞(Ω2)s2 → . . . , is algebraically exact, then the complex splits at C∞(Ωk)sk for k = 1, 2, . . . . It splits at C∞(Ω0)s iff ker T0 is strict graded.

Splitting of differential complexes

Fr´ echet spaces can be graded, i. e. written as projective limit in different ways: · n fundamental system of semi-norms in E.

• 1. En ‘Banach grading’,
• 2. E/ · n with quotient topology ‘strict norm grading’.

Ω ⊂ Rd open, K1 ⊂ K2 ⊂ .. compact exhaustion, E (Kn) is strict norm grading, Proj1/Ext1-method yields: Theorem: If Ωi ⊂ Rd are open subsets and T0 : C∞(Ω0)s → C∞(Ω1)s1 is a matrix of convolution operators such that the corresponding complex 0 → ker T0 → C∞(Ω0)s T0 → C∞(Ω1)s1 T1 → C∞(Ω2)s2 → . . . , is algebraically exact, then the complex splits at C∞(Ωk)sk for k = 1, 2, . . . . It splits at C∞(Ω0)s iff ker T0 is strict graded. Analogous result where C∞(Ω)s is replaced with D ′(Ω)s.

Hadamard operators

Ω ⊂ Rd open, E(Ω) = A(Ω) or C∞(Ω). Definition: A map L ∈ L(E(Ω)) is called a Hadamard operator (L ∈ M(E(Ω))) if it admits all monomials as eigenvectors.

Hadamard operators

Ω ⊂ Rd open, E(Ω) = A(Ω) or C∞(Ω). Definition: A map L ∈ L(E(Ω)) is called a Hadamard operator (L ∈ M(E(Ω))) if it admits all monomials as eigenvectors. ◮ L ∈ M(E(Ω)): L(xα) = mαxα. ◮ L → (mα)α∈Nd injective algebra homomorphism. ◮ ⇒ M(E(Ω)) is a closed commutative subalgebra of L(E(Ω)).

Hadamard operators

Ω ⊂ Rd open, E(Ω) = A(Ω) or C∞(Ω). Definition: A map L ∈ L(E(Ω)) is called a Hadamard operator (L ∈ M(E(Ω))) if it admits all monomials as eigenvectors. ◮ L ∈ M(E(Ω)): L(xα) = mαxα. ◮ L → (mα)α∈Nd injective algebra homomorphism. ◮ ⇒ M(E(Ω)) is a closed commutative subalgebra of L(E(Ω)). Representation theorem: Hadamard operators on E(Ω) have the form (MTf )(y) = Txf (xy), where T ∈ E ′, supp T ⊂ V (Ω). Notations: Multiplication xy = (x1y1, . . . , xdyd) V (Ω) = {x : xy ∈ Ω for all y ∈ Ω} ‘dilation set’

Hadamard operators

Ω ⊂ Rd open, E(Ω) = A(Ω) or C∞(Ω). Definition: A map L ∈ L(E(Ω)) is called a Hadamard operator (L ∈ M(E(Ω))) if it admits all monomials as eigenvectors. ◮ L ∈ M(E(Ω)): L(xα) = mαxα. ◮ L → (mα)α∈Nd injective algebra homomorphism. ◮ ⇒ M(E(Ω)) is a closed commutative subalgebra of L(E(Ω)). Representation theorem: Hadamard operators on E(Ω) have the form (MTf )(y) = Txf (xy), where T ∈ E ′, supp T ⊂ V (Ω). Notations: Multiplication xy = (x1y1, . . . , xdyd) V (Ω) = {x : xy ∈ Ω for all y ∈ Ω} ‘dilation set’ Consequences: 1. For L = MT: mα = Txα 2.MT ◦ MS = MT⋆S where (T ⋆ S)f = Tx(Syf (xy))

• 3. the Cauchy transform CT ∈ O0, CT⋆S = CT ⊙ CS where ⊙

denotes the Hadamard product.

Euler operators

Objects of study for Hadamard operators:

• M(E(Ω)) as topological algebra.
• Dilation sets.
• Global solvability of MTu = g.

Euler operators

Objects of study for Hadamard operators:

• M(E(Ω)) as topological algebra.
• Dilation sets.
• Global solvability of MTu = g.

Results on solvability for special case:

Euler partial differential operators:

P(θ) =

• α

cαθα, θj = xj∂/∂xj

Euler operators

Objects of study for Hadamard operators:

• M(E(Ω)) as topological algebra.
• Dilation sets.
• Global solvability of MTu = g.

Results on solvability for special case:

Euler partial differential operators:

P(θ) =

• α

cαθα, θj = xj∂/∂xj On Q =]0, +∞[d: P(θ)xz = P(z)xz for all z ∈ Cd, ⇒ mα = P(α). Mellin transform f (x), xz replaces Fourier transform.

Euler operators

Objects of study for Hadamard operators:

• M(E(Ω)) as topological algebra.
• Dilation sets.
• Global solvability of MTu = g.

Results on solvability for special case:

Euler partial differential operators:

P(θ) =

• α

cαθα, θj = xj∂/∂xj On Q =]0, +∞[d: P(θ)xz = P(z)xz for all z ∈ Cd, ⇒ mα = P(α). Mellin transform f (x), xz replaces Fourier transform. Problem: defined only on Q. Needs extended theory.

Global solutions

P polynomial, Pm principal part.

Global solutions

P polynomial, Pm principal part.

Case of E(Ω) = A(Ω), Ω = Rd

Necessary condition: P(θ)u = g ⇒ (*) ∂αg(0) = 0 if P(α) = 0.

Global solutions

P polynomial, Pm principal part.

Case of E(Ω) = A(Ω), Ω = Rd

Necessary condition: P(θ)u = g ⇒ (*) ∂αg(0) = 0 if P(α) = 0. Assume: Pm(ej) = 0 all j, ej unit vectors, P has finitely many integer zeros. Theorem: P(θ)u = g has a solution for all g ∈ A(Rd) with (*) ⇔ Pm(z) = 0 for all z ∈ Cd with Re z ∈ Q.

Global solutions

P polynomial, Pm principal part.

Case of E(Ω) = A(Ω), Ω = Rd

Necessary condition: P(θ)u = g ⇒ (*) ∂αg(0) = 0 if P(α) = 0. Assume: Pm(ej) = 0 all j, ej unit vectors, P has finitely many integer zeros. Theorem: P(θ)u = g has a solution for all g ∈ A(Rd) with (*) ⇔ Pm(z) = 0 for all z ∈ Cd with Re z ∈ Q.

Case of E(Ω) = C ∞(Ω), Ω = Rd

Necessary condition: P(θ)u = g ⇒ (**) For every α ∈ NJ, J ⊂ D = {1, .., d}: g(α)(0J, xD\J) = 0 if P(α, xD\J) = 0 on RD\J.

Global solutions

P polynomial, Pm principal part.

Case of E(Ω) = A(Ω), Ω = Rd

Necessary condition: P(θ)u = g ⇒ (*) ∂αg(0) = 0 if P(α) = 0. Assume: Pm(ej) = 0 all j, ej unit vectors, P has finitely many integer zeros. Theorem: P(θ)u = g has a solution for all g ∈ A(Rd) with (*) ⇔ Pm(z) = 0 for all z ∈ Cd with Re z ∈ Q.

Case of E(Ω) = C ∞(Ω), Ω = Rd

Necessary condition: P(θ)u = g ⇒ (**) For every α ∈ NJ, J ⊂ D = {1, .., d}: g(α)(0J, xD\J) = 0 if P(α, xD\J) = 0 on RD\J. Theorem: P(θ)u = g has a solution in C∞(Rd) for all g ∈ C∞(Rd) with (**).

Co-Authors of Pawe l Doma´ nski

Jos´ e Bonet Andreas Defant Susanne Dierolf Lech Drewnowski Juan Carlos Diaz Alcaide Carmen Fern´ andez Rosell Leonhard Frerick Michal Goli´ nski Bronis law Jakubczyk Michal Jasiczak Can Deha Kariksiz J¨

• rg Krone

Michael Langenbruch Mikael Lindstr¨

• m

Mieczys law Masty lo Jorge Mujica Augustyn Orty´ nski Melapalayam S. Ramanujan Georg Schl¨ uchtermann Jari Taskinen Dietmar Vogt Witold Wnuk