On surjectivity of partial differential operators with a single - - PowerPoint PPT Presentation

On surjectivity of partial differential operators with a single characteristic direction and on Runge pairs for such operators

Thomas Kalmes TU Chemnitz Dortmund-Hagen-Wuppertal Analysis Meeting TU Dortmund, January 24, 2019

Thomas Kalmes PDO with a single characteristic direction 1 / 25

1

Surjectivity of differential operators

2

An approximation theorem of Runge type

3

The linear topological invariant (Ω) for kernels

Thomas Kalmes PDO with a single characteristic direction 2 / 25

Surjectivity of differential operators

Thomas Kalmes PDO with a single characteristic direction 3 / 25

For P ∈ C[X1, . . . , Xd] set P(∂) := P(∂1, . . . , ∂d) := P( ∂ ∂x1 , . . . , ∂ ∂xd ). (E.g. ∆ = PL(∂) for PL(ξ) = d

j=1 ξ2 j (Laplace operator) ∂ ∂t − ∆x = PH(∂) for PH(ξ1, . . . , ξd) = ξd − d−1 j=1 ξ2 j (Heat operator)

i ∂

∂t + ∆x = PS(∂) for PS(ξ1, . . . , ξd) = iξd + d−1 j=1 ξ2 j (Schr¨

• dinger operator)

∂2 ∂t2 − ∆x = PW (∂) for PW (ξ1, . . . , ξd) = ξ2 d − d−1 j=1 ξ2 j (Wave operator) 1 2

∂

∂x1 + i ∂ ∂x2

• = ∂¯

z for P(ξ1, ξ2) = 1 2

• ξ1 + iξ2
• (Cauchy-Riemann operator).)

Thomas Kalmes PDO with a single characteristic direction 4 / 25

For P ∈ C[X1, . . . , Xd] set P(∂) := P(∂1, . . . , ∂d) := P( ∂ ∂x1 , . . . , ∂ ∂xd ). (E.g. ∆ = PL(∂) for PL(ξ) = d

j=1 ξ2 j (Laplace operator) ∂ ∂t − ∆x = PH(∂) for PH(ξ1, . . . , ξd) = ξd − d−1 j=1 ξ2 j (Heat operator)

i ∂

∂t + ∆x = PS(∂) for PS(ξ1, . . . , ξd) = iξd + d−1 j=1 ξ2 j (Schr¨

• dinger operator)

∂2 ∂t2 − ∆x = PW (∂) for PW (ξ1, . . . , ξd) = ξ2 d − d−1 j=1 ξ2 j (Wave operator) 1 2

∂

∂x1 + i ∂ ∂x2

• = ∂¯

z for P(ξ1, ξ2) = 1 2

• ξ1 + iξ2
• (Cauchy-Riemann operator).)

P ∈ C[X1, . . . , Xd]\{0}, X ⊆ Rd open: for given f, solve P(∂)u = f in X!

Thomas Kalmes PDO with a single characteristic direction 4 / 25

For P ∈ C[X1, . . . , Xd] set P(∂) := P(∂1, . . . , ∂d) := P( ∂ ∂x1 , . . . , ∂ ∂xd ). (E.g. ∆ = PL(∂) for PL(ξ) = d

j=1 ξ2 j (Laplace operator) ∂ ∂t − ∆x = PH(∂) for PH(ξ1, . . . , ξd) = ξd − d−1 j=1 ξ2 j (Heat operator)

i ∂

∂t + ∆x = PS(∂) for PS(ξ1, . . . , ξd) = iξd + d−1 j=1 ξ2 j (Schr¨

• dinger operator)

∂2 ∂t2 − ∆x = PW (∂) for PW (ξ1, . . . , ξd) = ξ2 d − d−1 j=1 ξ2 j (Wave operator) 1 2

∂

∂x1 + i ∂ ∂x2

• = ∂¯

z for P(ξ1, ξ2) = 1 2

• ξ1 + iξ2
• (Cauchy-Riemann operator).)

P ∈ C[X1, . . . , Xd]\{0}, X ⊆ Rd open: for given f, solve P(∂)u = f in X! Is this possible for every f from a fixed space of functions/distributions? ”Solution” in which sense; classical, distributional?

Thomas Kalmes PDO with a single characteristic direction 4 / 25

Example: X =

• (0, 2) × (−4, 4)
• ∪
• (−1, 1) × (−4, −2)
• ∪
• (−1, 1) × (2, 4)
• x2

x1 2 −1 −4 −3 −1 1 3 4

(1, x2) (x1, x2)

P1(ξ1, ξ2) = ξ1 ⇒ P1(∂) = ∂1; given f ∈ C∞(X) ⇒ u(x1, x2) := x1

1

f(t, x2) dt ∈ C∞(X) satisfies ∂1u = f ⇒ C∞(X) ⊆ P(∂)(C1(X))

Thomas Kalmes PDO with a single characteristic direction 5 / 25

Example: X =

• (0, 2) × (−4, 4)
• ∪
• (−1, 1) × (−4, −2)
• ∪
• (−1, 1) × (2, 4)
• x2

x1 2 −1 −4 −3 −1 1 3 4

(1, x2) (x1, x2)

P1(ξ1, ξ2) = ξ1 ⇒ P1(∂) = ∂1; given f ∈ C∞(X) ⇒ u(x1, x2) := x1

1

f(t, x2) dt ∈ C∞(X) satisfies ∂1u = f ⇒ P1(∂) : C∞(X) → C∞(X) surjective ⇒ C∞(X) ⊆ P(∂)(C1(X))

Thomas Kalmes PDO with a single characteristic direction 5 / 25

Example: X =

• (0, 2) × (−4, 4)
• ∪
• (−1, 1) × (−4, −2)
• ∪
• (−1, 1) × (2, 4)
• x2

x1 2 −1 −4 −3 −1 1 3 4

P2(ξ1, ξ2) = ξ2 ⇒ P2(∂) = ∂2; choose η ∈ C∞(R) with η(t) = 0 for t / ∈ [−1, 1] and 1

−1 η(t) dt > 0; set

f(x1, x2) = η(x2)

x1 ,

if x1 > 0 0, if x1 ≤ 0 ⇒ f ∈ C∞(X); assume ∃ u ∈ C1(X) : ∂2u = f; for x1 ∈ (0, 2) we then have u(x1, 3) − u(x1, −3) = 3

−3 ∂2u(x1, t)dt

=

1 x1

1

−1 η(t) dt →x1→0 ∞

⇒ C∞(X) ⊆ P(∂)(C1(X))

Thomas Kalmes PDO with a single characteristic direction 5 / 25

Example: X =

• (0, 2) × (−4, 4)
• ∪
• (−1, 1) × (−4, −2)
• ∪
• (−1, 1) × (2, 4)
• x2

x1 2 −1 −4 −3 −1 1 3 4

(x1, −3) (x1, 3)

P2(ξ1, ξ2) = ξ2 ⇒ P2(∂) = ∂2; choose η ∈ C∞(R) with η(t) = 0 for t / ∈ [−1, 1] and 1

−1 η(t) dt > 0; set

f(x1, x2) = η(x2)

x1 ,

if x1 > 0 0, if x1 ≤ 0 ⇒ f ∈ C∞(X); assume ∃ u ∈ C1(X) : ∂2u = f; for x1 ∈ (0, 2) we then have u(x1, 3) − u(x1, −3) = 3

−3 ∂2u(x1, t)dt

=

1 x1

1

−1 η(t) dt →x1→0 ∞

⇒ C∞(X) ⊆ P(∂)(C1(X))

Thomas Kalmes PDO with a single characteristic direction 5 / 25

Example: X =

• (0, 2) × (−4, 4)
• ∪
• (−1, 1) × (−4, −2)
• ∪
• (−1, 1) × (2, 4)
• x2

x1 2 −1 −4 −3 −1 1 3 4

(x1, −3) (x1, 3)

P2(ξ1, ξ2) = ξ2 ⇒ P2(∂) = ∂2; choose η ∈ C∞(R) with η(t) = 0 for t / ∈ [−1, 1] and 1

−1 η(t) dt > 0; set

f(x1, x2) = η(x2)

x1 ,

if x1 > 0 0, if x1 ≤ 0 ⇒ f ∈ C∞(X); assume ∃ u ∈ C1(X) : ∂2u = f; for x1 ∈ (0, 2) we then have u(x1, 3) − u(x1, −3) = 3

−3 ∂2u(x1, t)dt

=

1 x1

1

−1 η(t) dt →x1→0 ∞

⇒ C∞(X) ⊆ P(∂)(C1(X))

Thomas Kalmes PDO with a single characteristic direction 5 / 25

Example: X =

• (0, 2) × (−4, 4)
• ∪
• (−1, 1) × (−4, −2)
• ∪
• (−1, 1) × (2, 4)
• x2

x1 2 −1 −4 −3 −1 1 3 4

For P1(ξ1, ξ2) = ξ1 resp. P2(ξ1, ξ2) = ξ2 is P1(∂) : C∞(X) → C∞(X) surjective, P2(∂) : C∞(X) → C∞(X) not surjective. Is it possible to ”see” this without calculation? What about P2(∂) if we allow for more general solutions

• f P2(∂)u = f, f ∈ C∞(X), than

u ∈ C1(X)? ⇒ C∞(X) ⊆ P(∂)(C1(X))

Thomas Kalmes PDO with a single characteristic direction 5 / 25

Let P ∈ C[X1, . . . , Xd]\{0} and let X ⊆ Rd be open. i) When is P(∂) : C∞(X) → C∞(X) surjective? ii) When is C∞(X) ⊆ P(∂)(D′(X))?

Thomas Kalmes PDO with a single characteristic direction 6 / 25

Let P ∈ C[X1, . . . , Xd]\{0} and let X ⊆ Rd be open. i) When is P(∂) : C∞(X) → C∞(X) surjective? ii) When is C∞(X) ⊆ P(∂)(D′(X))?

• iii) When is P(∂) : D′(X) → D′(X) surjective?
• Thomas Kalmes

PDO with a single characteristic direction 6 / 25

Let P ∈ C[X1, . . . , Xd]\{0} and let X ⊆ Rd be open. i) When is P(∂) : C∞(X) → C∞(X) surjective? ii) When is C∞(X) ⊆ P(∂)(D′(X))?

• iii) When is P(∂) : D′(X) → D′(X) surjective?
• Answers will depend on combined properties of P and X.

Thomas Kalmes PDO with a single characteristic direction 6 / 25

Let P ∈ C[X1, . . . , Xd]\{0} and let X ⊆ Rd be open. i) When is P(∂) : C∞(X) → C∞(X) surjective? ii) When is C∞(X) ⊆ P(∂)(D′(X))?

• iii) When is P(∂) : D′(X) → D′(X) surjective?
• Answers will depend on combined properties of P and X.

Some thoughts on i): Equip C∞(X) with topology generated by the seminorms · l,K : C∞(X) → [0, ∞), f → max

α∈Nd

0,|α|≤l max

x∈K |∂αf(x)| (l ∈ N0, K ⋐ X)

⇒ C∞(X) Fr´ echet space, P(∂) : C∞(X) → C∞(X) continuous, linear.

Thomas Kalmes PDO with a single characteristic direction 6 / 25

Let P ∈ C[X1, . . . , Xd]\{0} and let X ⊆ Rd be open. i) When is P(∂) : C∞(X) → C∞(X) surjective? ii) When is C∞(X) ⊆ P(∂)(D′(X))?

• iii) When is P(∂) : D′(X) → D′(X) surjective?
• Answers will depend on combined properties of P and X.

Some thoughts on i): Equip C∞(X) with topology generated by the seminorms · l,K : C∞(X) → [0, ∞), f → max

α∈Nd

0,|α|≤l max

x∈K |∂αf(x)| (l ∈ N0, K ⋐ X)

⇒ C∞(X) Fr´ echet space, P(∂) : C∞(X) → C∞(X) continuous, linear. Abstract theory: F Fr´ echet, A : F → F continuous, linear A surjective ⇔ A has dense range, A has closed range

Thomas Kalmes PDO with a single characteristic direction 6 / 25

Let P ∈ C[X1, . . . , Xd]\{0} and let X ⊆ Rd be open. i) When is P(∂) : C∞(X) → C∞(X) surjective? ii) When is C∞(X) ⊆ P(∂)(D′(X))?

• iii) When is P(∂) : D′(X) → D′(X) surjective?
• Answers will depend on combined properties of P and X.

Some thoughts on i): Equip C∞(X) with topology generated by the seminorms · l,K : C∞(X) → [0, ∞), f → max

α∈Nd

0,|α|≤l max

x∈K |∂αf(x)| (l ∈ N0, K ⋐ X)

⇒ C∞(X) Fr´ echet space, P(∂) : C∞(X) → C∞(X) continuous, linear. Abstract theory: F Fr´ echet, A : F → F continuous, linear A surjective ⇔ A has dense range, A has closed range A has dense range ⇔ At : F ′ → F ′ injective (Hahn-Banach Theorem) A has closed range ⇔ At(F ′) is closed in (F ′, σ(F ′, F)) (Closed Range Theorem)

Thomas Kalmes PDO with a single characteristic direction 6 / 25

Let P ∈ C[X1, . . . , Xd]\{0} and let X ⊆ Rd be open. i) When is P(∂) : C∞(X) → C∞(X) surjective? ii) When is C∞(X) ⊆ P(∂)(D′(X))?

• iii) When is P(∂) : D′(X) → D′(X) surjective?
• Answers will depend on combined properties of P and X.

Some thoughts on i): Equip C∞(X) with topology generated by the seminorms · l,K : C∞(X) → [0, ∞), f → max

α∈Nd

0,|α|≤l max

x∈K |∂αf(x)| (l ∈ N0, K ⋐ X)

⇒ C∞(X) Fr´ echet space, P(∂) : C∞(X) → C∞(X) continuous, linear. Abstract theory: F Fr´ echet, A : F → F continuous, linear A surjective ⇔ A has dense range, A has closed range A has dense range ⇔ At : F ′ → F ′ injective (Hahn-Banach Theorem) A has closed range ⇔ At(F ′) is closed in (F ′, σ(F ′, F)) (Closed Range Theorem) C∞(X)′ = E ′(X) and P(∂)t = ˇ P(∂), where ˇ P(ξ) = P(−ξ). P(∂) : C∞(X) → C∞(X) has always dense range

Thomas Kalmes PDO with a single characteristic direction 6 / 25

Theorem (Malgrange, 1956)

Tfae: i) P(∂) : C∞(X) → C∞(X) is surjective. ii) X is P-convex for supports, i.e. ∀ K ⋐ X ∃ ˜ K ⋐ X ∀ u ∈ E ′(X) :

• supp ˇ

P(∂)u ⊆ K ⇒ supp u ⊆ ˜ K.

• Thomas Kalmes

PDO with a single characteristic direction 7 / 25

Theorem (Malgrange, 1956)

Tfae: i) P(∂) : C∞(X) → C∞(X) is surjective. ii) X is P-convex for supports, i.e. ∀ K ⋐ X ∃ ˜ K ⋐ X ∀ u ∈ E ′(X) :

• supp ˇ

P(∂)u ⊆ K ⇒ supp u ⊆ ˜ K.

• ⇔

∀ u ∈ E ′(X) : dist(Rd\X, supp u) ≥ dist(Rd\X, supp ˇ P(∂)u) ⇔ ∀ ϕ ∈ D(X) : dist(Rd\X, supp ϕ) ≥ dist(Rd\X, supp ˇ P(∂)ϕ)

Thomas Kalmes PDO with a single characteristic direction 7 / 25

Theorem (Malgrange, 1956)

Tfae: i) P(∂) : C∞(X) → C∞(X) is surjective. ii) X is P-convex for supports, i.e. ∀ K ⋐ X ∃ ˜ K ⋐ X ∀ u ∈ E ′(X) :

• supp ˇ

P(∂)u ⊆ K ⇒ supp u ⊆ ˜ K.

• ⇔

∀ u ∈ E ′(X) : dist(Rd\X, supp u) ≥ dist(Rd\X, supp ˇ P(∂)u) ⇔ ∀ ϕ ∈ D(X) : dist(Rd\X, supp ϕ) ≥ dist(Rd\X, supp ˇ P(∂)ϕ) iii) C∞(X) ⊆ P(∂)

• D′(X)
• .

Thomas Kalmes PDO with a single characteristic direction 7 / 25

Theorem (Malgrange, 1956)

Tfae: i) P(∂) : C∞(X) → C∞(X) is surjective. ii) X is P-convex for supports, i.e. ∀ K ⋐ X ∃ ˜ K ⋐ X ∀ u ∈ E ′(X) :

• supp ˇ

P(∂)u ⊆ K ⇒ supp u ⊆ ˜ K.

• ⇔

∀ u ∈ E ′(X) : dist(Rd\X, supp u) ≥ dist(Rd\X, supp ˇ P(∂)u) ⇔ ∀ ϕ ∈ D(X) : dist(Rd\X, supp ϕ) ≥ dist(Rd\X, supp ˇ P(∂)ϕ) iii) C∞(X) ⊆ P(∂)

• D′(X)
• .

iv) P(∂) : D′

F (X) → D′ F (X) is surjective.

v) For all s ∈ R it holds Hs,loc(X) ⊆ P(∂)

• Hs,loc(X)
• .

Thomas Kalmes PDO with a single characteristic direction 7 / 25

Theorem (Malgrange, 1956)

Tfae: i) P(∂) : C∞(X) → C∞(X) is surjective. ii) X is P-convex for supports, i.e. ∀ K ⋐ X ∃ ˜ K ⋐ X ∀ u ∈ E ′(X) :

• supp ˇ

P(∂)u ⊆ K ⇒ supp u ⊆ ˜ K.

• ⇔

∀ u ∈ E ′(X) : dist(Rd\X, supp u) ≥ dist(Rd\X, supp ˇ P(∂)u) ⇔ ∀ ϕ ∈ D(X) : dist(Rd\X, supp ϕ) ≥ dist(Rd\X, supp ˇ P(∂)ϕ) iii) C∞(X) ⊆ P(∂)

• D′(X)
• .

iv) P(∂) : D′

F (X) → D′ F (X) is surjective.

v) For all s ∈ R it holds Hs,loc(X) ⊆ P(∂)

• Hs,loc(X)
• .

Theorem of supports: ∀ u ∈ E ′(X) : conv

• supp u
• = conv
• supp ˇ

P(∂)u

• ⇒ every convex X is P-convex for supports (Recall: P = 0!)

Thomas Kalmes PDO with a single characteristic direction 7 / 25

Geometrical conditions for/characterization of P-convexity for supports? Problem: not a local property!

Thomas Kalmes PDO with a single characteristic direction 8 / 25

Geometrical conditions for/characterization of P-convexity for supports? Problem: not a local property! Every open X ⊆ Rd is P-convex for supports iff P is elliptic, i.e. if P(ξ) =

|α|≤m aαξα then

∀ ξ ∈ Rd\{0}; 0 = Pm(ξ) :=

• |α|=m

aαξα (principal part of P)

Thomas Kalmes PDO with a single characteristic direction 8 / 25

Geometrical conditions for/characterization of P-convexity for supports? Problem: not a local property! Every open X ⊆ Rd is P-convex for supports iff P is elliptic, i.e. if P(ξ) =

|α|≤m aαξα then

∀ ξ ∈ Rd\{0}; 0 = Pm(ξ) :=

• |α|=m

aαξα (principal part of P) If P acts along a subspace of Rd and is elliptic there, then P-convexity for supports is completely characterized (Nakane, 1979). For polynomials with principal part P2(ξ) = ξ2

d − d−1 j=1 ξ2 j P-convexity for

supports is completely characterized (Persson, 1981). For P of real principal type there are characterizations if

• X is bounded and ∂X is analytic (Tintarev, 1988)
• X ⊆ R3 (Tintarev, 1992)

Thomas Kalmes PDO with a single characteristic direction 8 / 25

Geometrical conditions for/characterization of P-convexity for supports? Problem: not a local property! Every open X ⊆ Rd is P-convex for supports iff P is elliptic, i.e. if P(ξ) =

|α|≤m aαξα then

∀ ξ ∈ Rd\{0}; 0 = Pm(ξ) :=

• |α|=m

aαξα (principal part of P) If P acts along a subspace of Rd and is elliptic there, then P-convexity for supports is completely characterized (Nakane, 1979). For polynomials with principal part P2(ξ) = ξ2

d − d−1 j=1 ξ2 j P-convexity for

supports is completely characterized (Persson, 1981). For P of real principal type there are characterizations if

• X is bounded and ∂X is analytic (Tintarev, 1988)
• X ⊆ R3 (Tintarev, 1992)

For d = 2 P-convexity for supports is completely characterized (H¨

• rmander,

1971).

Thomas Kalmes PDO with a single characteristic direction 8 / 25

Necessary condition for P-convexity for supports: f : X → R satisfies the minimum principle in a (fixed) closed subset F of Rd if for every compact set K ⊆ F ∩ X we have inf

x∈K f(x) = inf ∂F K f(x),

where ∂F K denotes the boundary of K in F.

Thomas Kalmes PDO with a single characteristic direction 9 / 25

Necessary condition for P-convexity for supports: f : X → R satisfies the minimum principle in a (fixed) closed subset F of Rd if for every compact set K ⊆ F ∩ X we have inf

x∈K f(x) = inf ∂F K f(x),

where ∂F K denotes the boundary of K in F. We set dX : X → R, x → dist(x, Rd\X), the boundary distance of X. X is P-convex for supports ⇒ dX satisfies the minimum principle in every characteristic hyperplane H for P, i.e. in H = x +

• span{N}

⊥ (x ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0). span{N} characteristic direction of P

Thomas Kalmes PDO with a single characteristic direction 9 / 25

Necessary condition for P-convexity for supports: f : X → R satisfies the minimum principle in a (fixed) closed subset F of Rd if for every compact set K ⊆ F ∩ X we have inf

x∈K f(x) = inf ∂F K f(x),

where ∂F K denotes the boundary of K in F. We set dX : X → R, x → dist(x, Rd\X), the boundary distance of X. X is P-convex for supports ⇒ dX satisfies the minimum principle in every characteristic hyperplane H for P, i.e. in H = x +

• span{N}

⊥ (x ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0). span{N} characteristic direction of P

x2 x1

• min. principle satisfied in
• char. for P (∂) = ∂1

x2 x1

• min. principle not satisfied in
• char. for P (∂) = ∂2

K Thomas Kalmes PDO with a single characteristic direction 9 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports.

Thomas Kalmes PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports. Sketch of proof: X P-convex for supports ⇔ ∀ u ∈ E ′(X) : dist(supp u, Rd\X) ≥ dist(supp ˇ P(∂)u, Rd\X) Set K := supp ˇ P(∂)u and fix x ∈ {y ∈ X; dist(y, Rd\X) < dist(K, Rd\X)}, we have to show x / ∈ supp u. Since dX satisfies the minimum principle in x + W ⊥ one can show the existence of α : [0, T] → X2 cont. piecewise affine: α(0) = x, α([0, T]) ∩ K = ∅, α(T) / ∈ supp u, α([0, T]) ⊆ x + W ⊥, By α2) + α3) ∃ ε > 0 : supp u ∩ B(α(T), ε) = ∅, K ∩

• α([0, T]) + B(0, ε)
• = ∅.

By α4) : ∀ y ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0 :

• Hy,N ∩
• α([0, T]) + B(0, ε)
• = ∅ ⇒ Hy,N ∩ B(α(T), ε) = ∅
• .

With Holmgren’s uniqueness theorem [H¨

• rmander’s continuation of differentiability

theorem]: supp u ∩ B(α(0), ε) = ∅, thus by α1) x / ∈ supp u

• Thomas Kalmes

PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports. Sketch of proof: X P-convex for supports ⇔ ∀ u ∈ E ′(X) : dist(supp u, Rd\X) ≥ dist(supp ˇ P(∂)u, Rd\X) Set K := supp ˇ P(∂)u and fix x ∈ {y ∈ X; dist(y, Rd\X) < dist(K, Rd\X)}, we have to show x / ∈ supp u. Since dX satisfies the minimum principle in x + W ⊥ one can show the existence of α : [0, T] → X cont. piecewise affine: α(0) = x, α([0, T]) ∩ K = ∅, α(T) / ∈ supp u, α([0, T]) ⊆ x + W ⊥, By α2) + α3) ∃ ε > 0 : supp u ∩ B(α(T), ε) = ∅, K ∩

• α([0, T]) + B(0, ε)
• = ∅.

By α4) : ∀ y ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0 :

• Hy,N ∩
• α([0, T]) + B(0, ε)
• = ∅ ⇒ Hy,N ∩ B(α(T), ε) = ∅
• .

With Holmgren’s uniqueness theorem [H¨

• rmander’s continuation of differentiability

theorem]: supp u ∩ B(α(0), ε) = ∅, thus by α1) x / ∈ supp u

• Thomas Kalmes

PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports. Sketch of proof: X P-convex for supports ⇔ ∀ u ∈ E ′(X) : dist(supp u, Rd\X) ≥ dist(supp ˇ P(∂)u, Rd\X) Set K := supp ˇ P(∂)u and fix x ∈ {y ∈ X; dist(y, Rd\X) < dist(K, Rd\X)}, we have to show x / ∈ supp u. Since dX satisfies the minimum principle in x + W ⊥ one can show the existence of α : [0, T] → X cont. piecewise affine: α1) α(0) = x, α2) α([0, T]) ∩ K = ∅, α3) α(T) / ∈ supp u, α4) α([0, T]) ⊆ x + W ⊥, By α2) + α3) ∃ ε > 0 : supp u ∩ B(α(T), ε) = ∅, K ∩

• α([0, T]) + B(0, ε)
• = ∅.

By α4) : ∀ y ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0 :

• Hy,N ∩
• α([0, T]) + B(0, ε)
• = ∅ ⇒ Hy,N ∩ B(α(T), ε) = ∅
• .

With Holmgren’s uniqueness theorem [H¨

• rmander’s continuation of differentiability

theorem]: supp u ∩ B(α(0), ε) = ∅, thus by α1) x / ∈ supp u

• Thomas Kalmes

PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports. Sketch of proof: X P-convex for supports ⇔ ∀ u ∈ E ′(X) : dist(supp u, Rd\X) ≥ dist(supp ˇ P(∂)u, Rd\X) Set K := supp ˇ P(∂)u and fix x ∈ {y ∈ X; dist(y, Rd\X) < dist(K, Rd\X)}, we have to show x / ∈ supp u. Since dX satisfies the minimum principle in x + W ⊥ one can show the existence of α : [0, T] → X cont. piecewise affine: α1) α(0) = x, α2) α([0, T]) ∩ K = ∅, α3) α(T) / ∈ supp u, α4) α([0, T]) ⊆ x + W ⊥, By α2) + α3) ∃ ε > 0 : supp u ∩ B(α(T), ε) = ∅, K ∩

• α([0, T]) + B(0, ε)
• = ∅.

By α4) : ∀ y ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0 :

• Hy,N ∩
• α([0, T]) + B(0, ε)
• = ∅ ⇒ Hy,N ∩ B(α(T), ε) = ∅
• .

With Holmgren’s uniqueness theorem [H¨

• rmander’s continuation of differentiability

theorem]: supp u ∩ B(α(0), ε) = ∅, thus by α1) x / ∈ supp u

• Thomas Kalmes

PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports. Sketch of proof: X P-convex for supports ⇔ ∀ u ∈ E ′(X) : dist(supp u, Rd\X) ≥ dist(supp ˇ P(∂)u, Rd\X) Set K := supp ˇ P(∂)u and fix x ∈ {y ∈ X; dist(y, Rd\X) < dist(K, Rd\X)}, we have to show x / ∈ supp u. Since dX satisfies the minimum principle in x + W ⊥ one can show the existence of α : [0, T] → X cont. piecewise affine: α1) α(0) = x, α2) α([0, T]) ∩ K = ∅, α3) α(T) / ∈ supp u, α4) α([0, T]) ⊆ x + W ⊥, By α2) + α3) ∃ ε > 0 : supp u ∩ B(α(T), ε) = ∅, K ∩

• α([0, T]) + B(0, ε)
• = ∅.

By α4) : ∀ y ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0 :

• Hy,N ∩
• α([0, T]) + B(0, ε)
• = ∅ ⇒ Hy,N ∩ B(α(T), ε) = ∅
• .

where Hy,N := y +

• span{N}

⊥[H¨

• rmander’s continuation of differentiability the-
• rem]: supp u ∩ B(α(0), ε) = ∅, thus by α1) x /

∈ supp u

• Thomas Kalmes

PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports. Sketch of proof: X P-convex for supports ⇔ ∀ u ∈ E ′(X) : dist(supp u, Rd\X) ≥ dist(supp ˇ P(∂)u, Rd\X) Set K := supp ˇ P(∂)u and fix x ∈ {y ∈ X; dist(y, Rd\X) < dist(K, Rd\X)}, we have to show x / ∈ supp u. Since dX satisfies the minimum principle in x + W ⊥ one can show the existence of α : [0, T] → X cont. piecewise affine: α1) α(0) = x, α2) α([0, T]) ∩ K = ∅, α3) α(T) / ∈ supp u, α4) α([0, T]) ⊆ x + W ⊥, By α2) + α3) ∃ ε > 0 : supp u ∩ B(α(T), ε) = ∅, K ∩

• α([0, T]) + B(0, ε)
• = ∅.

By α4) : ∀ y ∈ Rd, N ∈ Rd, |N| = 1, Pm(N) = 0 :

• Hy,N ∩
• α([0, T]) + B(0, ε)
• = ∅ ⇒ Hy,N ∩ B(α(T), ε) = ∅
• .

u|B(α(T ),ε) = 0 and a UCP now imply u|B(α(0),ε) = 0, thus by α1) x / ∈ supp u

• Thomas Kalmes

PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports.

Thomas Kalmes PDO with a single characteristic direction 10 / 25

Theorem [6, Theorem 1]

Given P with principal part Pm such that {ξ ∈ Rd; Pm(ξ) = 0} ⊆ W for a subspace W Rd. Moreover, let X ⊆ Rd be open such that dX satisfies the minimum principle in x + W ⊥ for every x ∈ Rd. Then X is P-convex for supports.

Corollary [6, Corollary 5]

i) Given P with principal part Pm, {ξ ∈ Rd; Pm(ξ) = 0} = span{N}, |N| = 1. Then X is P-convex for supports if and only if dX satisfies the minimum principle in x +

• span{N}

⊥ for every x ∈ Rd. ii) Let p ∈ N0, d ∈ N, d ≥ 2, and let Q ∈ C[X1, . . . , Xd−1] be elliptic with deg(Q) =: m ≥ p + 1. Moreover, let Y ⊆ Rd−1, I ⊆ R be open. The

• perator

∂p ∂tp − Q(∂y) : C∞(Y × I) → C∞(Y × I) is surjective (coefficients of Qm real, p = 1: non-degenerate parabolic

• perator; more general, p odd: p-parabolic operator).

Thomas Kalmes PDO with a single characteristic direction 10 / 25

An approximation theorem of Runge type

Thomas Kalmes PDO with a single characteristic direction 11 / 25

Runge’s Approximation Theorem

For X1 ⊆ X2 ⊆ C open the following are equivalent. i) For every g ∈ H (X1), for every compact K ⊆ X1, and for every ε > 0 there is f ∈ H (X2) such that ε > sup

z∈K

|f(z) − g(z)| = f − g0,K, i.e. r : H (X2) → H (X1), f → f|X1 has dense range when H (X1) is equipped with the compact-open topology (topology of local uniform convergence); (X1, X2) is a Runge pair. ii) For every compact connected component C of C\X1 it holds C ⊆ X2.

Thomas Kalmes PDO with a single characteristic direction 12 / 25

For P ∈ C[X1, . . . , Xd]\{0} set C∞

P (X) := {f ∈ C∞(X); P(∂)f = 0 in X}.

⇒ C∞

P (X) is a closed subspace of C∞(X) thus a Fr´

echet space.

Thomas Kalmes PDO with a single characteristic direction 13 / 25

For P ∈ C[X1, . . . , Xd]\{0} set C∞

P (X) := {f ∈ C∞(X); P(∂)f = 0 in X}.

⇒ C∞

P (X) is a closed subspace of C∞(X) thus a Fr´

echet space. Set D′

P (X) := {u ∈ D′(X); P(∂)u = 0}.

P hypoelliptic :⇔ ∀ X ⊆ Rd open : D′

P (X) = C∞ P (X)

P elliptic ⇒ P hypoelliptic

Thomas Kalmes PDO with a single characteristic direction 13 / 25

For P ∈ C[X1, . . . , Xd]\{0} set C∞

P (X) := {f ∈ C∞(X); P(∂)f = 0 in X}.

⇒ C∞

P (X) is a closed subspace of C∞(X) thus a Fr´

echet space. Set D′

P (X) := {u ∈ D′(X); P(∂)u = 0}.

P hypoelliptic :⇔ ∀ X ⊆ Rd open : D′

P (X) = C∞ P (X)

P elliptic ⇒ P hypoelliptic We equip D′(X) with the strong dual topology and endow D′

P (X) with the

subspace topology. P hypoelliptic ⇒ C∞

P (X) = D′ P (X) as locally convex spaces and therefore:

topology of C∞

P (X) is generated by the seminorms { · 0,K; K ⋐ X}, i.e. it is

the compact-open topology (topology of local uniform convergence).

Thomas Kalmes PDO with a single characteristic direction 13 / 25

For P ∈ C[X1, . . . , Xd]\{0} set C∞

P (X) := {f ∈ C∞(X); P(∂)f = 0 in X}.

⇒ C∞

P (X) is a closed subspace of C∞(X) thus a Fr´

echet space. Set D′

P (X) := {u ∈ D′(X); P(∂)u = 0}.

P hypoelliptic :⇔ ∀ X ⊆ Rd open : D′

P (X) = C∞ P (X)

P elliptic ⇒ P hypoelliptic We equip D′(X) with the strong dual topology and endow D′

P (X) with the

subspace topology. P hypoelliptic ⇒ C∞

P (X) = D′ P (X) as locally convex spaces and therefore:

topology of C∞

P (X) is generated by the seminorms { · 0,K; K ⋐ X}, i.e. it is

the compact-open topology (topology of local uniform convergence). d = 2, P(ξ1, ξ2) = 1

2(ξ1 + iξ2) ⇒ C∞ P (X) = H (X), X ⊆ C = R2 open

Thomas Kalmes PDO with a single characteristic direction 13 / 25

Lax-Malgrange Theorem

For X1 ⊆ X2 ⊆ Rd open and P elliptic the following are equivalent. i) The restriction map rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has dense range,

i.e. (X1, X2) is a Runge pair for P(∂). ii) For every compact connected component C of Rd\X1 it holds C ⊆ X2.

Thomas Kalmes PDO with a single characteristic direction 14 / 25

Lax-Malgrange Theorem

For X1 ⊆ X2 ⊆ Rd open and P elliptic the following are equivalent. i) The restriction map rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has dense range,

i.e. (X1, X2) is a Runge pair for P(∂). ii) For every compact connected component C of Rd\X1 it holds C ⊆ X2. d = 2, P(∂) = ∂¯

z = 1 2(∂1 + i∂2) gives Runge’s Approximation Theorem.

Thomas Kalmes PDO with a single characteristic direction 14 / 25

Lax-Malgrange Theorem

For X1 ⊆ X2 ⊆ Rd open and P elliptic the following are equivalent. i) The restriction map rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has dense range,

i.e. (X1, X2) is a Runge pair for P(∂). ii) For every compact connected component C of Rd\X1 it holds C ⊆ X2. d = 2, P(∂) = ∂¯

z = 1 2(∂1 + i∂2) gives Runge’s Approximation Theorem.

Consider the class of differential operators P(∂) for which ∃ N ∈ Rd, |N| = 1 : {ξ ∈ Rd; Pm(ξ) = 0} = span{N} which contains e.g. P(∂) = ∂p

∂tp − Q(∂y) for x = (y, t) = (y1, . . . , yd−1, t) ∈ Rd

where Q ∈ C[X1, . . . , Xd−1] is elliptic with deg(Q) =: m ≥ p + 1 (heat operator, time dependent free Schr¨

• dinger operator, etc.).

When have rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1, resp. rD′ : D′ P (X2) → D′ P (X1), u → u|X1,

dense range?

Thomas Kalmes PDO with a single characteristic direction 14 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1.

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. For elliptic P the above yields Lax-Malgrange: ∃ C compact connected component of Rd\X1, C ⊆ X2 ⇒ ∃ψ ∈ D(X1 ∪ C); ψ = 1 in neighborhood of C so for ζ ∈ Cd, ˇ P(ζ) = 0 supp ˇ P(∂)(eζ,·ψ) ⊆ X1. Thus, iv) does not hold for ϕ := eζ,·ψ

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. For elliptic P the above yields Lax-Malgrange: ∃ C compact connected component of Rd\X1, C ⊆ X2 ⇒ ∃ψ ∈ D(X1 ∪ C); ψ = 1 in neighborhood of C so for ζ ∈ Cd, ˇ P(ζ) = 0 supp ˇ P(∂)(eζ,·ψ) ⊆ X1. Thus, iv) does not hold for ϕ := eζ,·ψ Assume, no compact connected component of Rd\X1 is contained in X2. Given ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. For elliptic P the above yields Lax-Malgrange: ∃ C compact connected component of Rd\X1, C ⊆ X2 ⇒ ∃ψ ∈ D(X1 ∪ C); ψ = 1 in neighborhood of C so for ζ ∈ Cd, ˇ P(ζ) = 0 supp ˇ P(∂)(eζ,·ψ) ⊆ X1. Thus, iv) does not hold for ϕ := eζ,·ψ Assume, no compact connected component of Rd\X1 is contained in X2. Given ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 ⇒ ϕ real analytic in Rd\X1

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. For elliptic P the above yields Lax-Malgrange: ∃ C compact connected component of Rd\X1, C ⊆ X2 ⇒ ∃ψ ∈ D(X1 ∪ C); ψ = 1 in neighborhood of C so for ζ ∈ Cd, ˇ P(ζ) = 0 supp ˇ P(∂)(eζ,·ψ) ⊆ X1. Thus, iv) does not hold for ϕ := eζ,·ψ Assume, no compact connected component of Rd\X1 is contained in X2. Given ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 ⇒ ϕ real analytic in Rd\X1 By hypothesis, every connected component of Rd\X1 intersects ∂∞X2 (boundary of X2 in the one-point compactification of Rd) and ϕ = 0 in a neighborhood of ∂∞X2

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. For elliptic P the above yields Lax-Malgrange: ∃ C compact connected component of Rd\X1, C ⊆ X2 ⇒ ∃ψ ∈ D(X1 ∪ C); ψ = 1 in neighborhood of C so for ζ ∈ Cd, ˇ P(ζ) = 0 supp ˇ P(∂)(eζ,·ψ) ⊆ X1. Thus, iv) does not hold for ϕ := eζ,·ψ Assume, no compact connected component of Rd\X1 is contained in X2. Given ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 ⇒ ϕ real analytic in Rd\X1 By hypothesis, every connected component of Rd\X1 intersects ∂∞X2 (boundary of X2 in the one-point compactification of Rd) and ϕ = 0 in a neighborhood of ∂∞X2 ⇒ ϕ|Rd\X1 = 0, i.e. iv) holds.

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1.

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. With Hahn-Banach Theorem: rC∞, resp. rD′, has dense range iff rt

C∞, resp. rt D′, is injective.

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. With Hahn-Banach Theorem: rC∞, resp. rD′, has dense range iff rt

C∞, resp. rt D′, is injective.

Hence, a representation of C∞

P (Xj)′, resp. D′ P (Xj)′, will be useful for the proof.

Thomas Kalmes PDO with a single characteristic direction 15 / 25

Representation of C∞

P (X)′ for X being P-convex for supports due to

Grothendieck: Fix fundamental solution E for ˇ P(∂). For K ⋐ Rd we call u ∈ D′

ˇ P (Rd\K) regular at infinity w.r.t. E iff for one

(then every) ψ ∈ C∞(Rd) with supp ψ ∩ K = ∅ and supp (1 − ψ) compact: E ∗ ˇ P(∂)(ψu) = ψu. R ˇ

P (Rd\K) := {u ∈ D′ ˇ P (Rd\K); u regular w.r.t. E}

Thomas Kalmes PDO with a single characteristic direction 16 / 25

Representation of C∞

P (X)′ for X being P-convex for supports due to

Grothendieck: Fix fundamental solution E for ˇ P(∂). For K ⋐ Rd we call u ∈ D′

ˇ P (Rd\K) regular at infinity w.r.t. E iff for one

(then every) ψ ∈ C∞(Rd) with supp ψ ∩ K = ∅ and supp (1 − ψ) compact: E ∗ ˇ P(∂)(ψu) = ψu. R ˇ

P (Rd\K) := {u ∈ D′ ˇ P (Rd\K); u regular w.r.t. E}

R ˇ

P (Xc) := ∪K⋐XR ˇ P (Rd\K). For u, v ∈ R ˇ P (Xc) we define

u ∼ v :⇔ ∃ L ⋐ X : u|Rd\L = v|Rd\L

Thomas Kalmes PDO with a single characteristic direction 16 / 25

Representation of C∞

P (X)′ for X being P-convex for supports due to

Grothendieck: Fix fundamental solution E for ˇ P(∂). For K ⋐ Rd we call u ∈ D′

ˇ P (Rd\K) regular at infinity w.r.t. E iff for one

(then every) ψ ∈ C∞(Rd) with supp ψ ∩ K = ∅ and supp (1 − ψ) compact: E ∗ ˇ P(∂)(ψu) = ψu. R ˇ

P (Rd\K) := {u ∈ D′ ˇ P (Rd\K); u regular w.r.t. E}

R ˇ

P (Xc) := ∪K⋐XR ˇ P (Rd\K). For u, v ∈ R ˇ P (Xc) we define

u ∼ v :⇔ ∃ L ⋐ X : u|Rd\L = v|Rd\L ⇒ ΦX : R ˇ

P (Xc)/∼ → C∞ P (X)′, ΦX([u]∼), f := ˇ

P(∂)(ψu), f well-defined (topological) isomorphism (ψ is as above, supp (1 − ψ) ⊆ X)

Thomas Kalmes PDO with a single characteristic direction 16 / 25

Representation of C∞

P (X)′ for X being P-convex for supports due to

Grothendieck: Fix fundamental solution E for ˇ P(∂). For K ⋐ Rd we call u ∈ D′

ˇ P (Rd\K) regular at infinity w.r.t. E iff for one

(then every) ψ ∈ C∞(Rd) with supp ψ ∩ K = ∅ and supp (1 − ψ) compact: E ∗ ˇ P(∂)(ψu) = ψu. R ˇ

P (Rd\K) := {u ∈ D′ ˇ P (Rd\K); u regular w.r.t. E}

R ˇ

P (Xc) := ∪K⋐XR ˇ P (Rd\K). For u, v ∈ R ˇ P (Xc) we define

u ∼ v :⇔ ∃ L ⋐ X : u|Rd\L = v|Rd\L ⇒ ΦX : R ˇ

P (Xc)/∼ → C∞ P (X)′, ΦX([u]∼), f := ˇ

P(∂)(ψu), f well-defined (topological) isomorphism (ψ is as above, supp (1 − ψ) ⊆ X) Example (K¨

• the, 1953):

P(∂) = ∂¯

z = 1 2(∂1 + i∂2), X = B(0, 1) ⊆ R2 = C, E(z) = 1 πz.

⇒ R ˇ

P (B(0, 1)c) = {u ∈ H (C\B(0, 1)); lim|z|→∞ u(z) = 0}

∀ f, u : Φ([u]∼), f = −

• C

∂¯

z(ψu)(z)f(z)dz =

1 2πi

• |z|=1− ε

2

u(z)f(z) dz. where ε ∈ (0, 1) is such that u has a holomorphic representative on C\B[0, 1 − ε].

Thomas Kalmes PDO with a single characteristic direction 16 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”:

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: P(∂) : C∞(X1) → C∞(X1) has dense range. By iii) and P(∂)t = ˇ P(∂) it follows that P(∂)t(E ′(X1)) is closed in E ′(X1). By the Closed Range Theorem for Fr´ echet spaces P(∂)(C∞(X1)) is closed in C∞(X1). Thus, P(∂) : C∞(X1) → C∞(X1) is surjective, i.e. X1 is P-convex for supports.

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: P(∂) : C∞(X1) → C∞(X1) has dense range. By iii) and P(∂)t = ˇ P(∂) it follows that P(∂)t(E ′(X1)) is closed in E ′(X1). By the Closed Range Theorem for Fr´ echet spaces P(∂)(C∞(X1)) is closed in C∞(X1). Thus, P(∂) : C∞(X1) → C∞(X1) is surjective, i.e. X1 is P-convex for supports.

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: iii) implies that X1 is P-convex for supports.

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: iii) implies that X1 is P-convex for supports. With Grothendieck duality: rt

C∞ injective iff

∀ u ∈ R ˇ

P (Xc 1) :

• rt

C∞(ΦX1([u]∼)) = 0 ⇒ u = 0 outside a compact subset of X1

• Let K ⋐ X1, u ∈ R ˇ

P (Rd\K) with rt C∞(ΦX1([u]∼)) = 0, i.e.

∀ f ∈ C∞

P (X2) : 0 = rt C∞(ΦX1([u]∼)), fX2 = ˇ

P(∂)(ψu), f|X1X1 = ˇ P(∂)(ψu), fX2 = ΦX2([u]∼), fX2, i.e. ΦX2([u]∼) = 0 where ψ ∈ C∞(Rd), supp ψ∩K = ∅ and supp (1−ψ) ⊆ X1 compact. ΦX2([u]∼) = 0 ⇒ ∃ L ⋐ X2 : ψu ∈ E ′(L), ˇ P(∂)(ψu) ∈ E ′(X1)

iii)

⇒ ψu ∈ E ′(X1)

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: iii) implies that X1 is P-convex for supports. With Grothendieck duality: rt

C∞ injective iff

∀ u ∈ R ˇ

P (Xc 1) :

• rt

C∞(ΦX1([u]∼)) = 0 ⇒ u = 0 outside a compact subset of X1

• Let K ⋐ X1, u ∈ R ˇ

P (Rd\K) with rt C∞(ΦX1([u]∼)) = 0, i.e.

∀ f ∈ C∞

P (X2) : 0 = rt C∞(ΦX1([u]∼)), fX2 = ˇ

P(∂)(ψu), f|X1X1 = ˇ P(∂)(ψu), fX2 = ΦX2([u]∼), fX2, i.e. ΦX2([u]∼) = 0 where ψ ∈ C∞(Rd), supp ψ ∩ K = ∅, supp (1 − ψ) ⊆ X1 compact. ΦX2([u]∼) = 0 ⇒ ∃ L ⊆ X2 compact : ψu ∈ E ′(L), ˇ P(∂)(ψu) ∈ E ′(X1)

iii)

⇒ ψu ∈ E ′(X1)

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: iii) implies that X1 is P-convex for supports. With Grothendieck duality: rt

C∞ injective iff

∀ u ∈ R ˇ

P (Xc 1) :

• rt

C∞(ΦX1([u]∼)) = 0 ⇒ u = 0 outside a compact subset of X1

• Let K ⋐ X1, u ∈ R ˇ

P (Rd\K) with rt C∞(ΦX1([u]∼)) = 0, i.e.

∀ f ∈ C∞

P (X2) : 0 = rt C∞(ΦX1([u]∼)), fX2 = ˇ

P(∂)(ψu), f|X1X1 = ˇ P(∂)(ψu), fX2 = ΦX2([u]∼), fX2, i.e. ΦX2([u]∼) = 0 where ψ ∈ C∞(Rd), supp ψ ∩ K = ∅, supp (1 − ψ) ⊆ X1 compact. ΦX2([u]∼) = 0 ⇒ ∃ L ⋐ X2 : ψu ∈ E ′(L), ˇ P(∂)(ψu) ∈ E ′(X1)

iii)

⇒ ψu ∈ E ′(X1)

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: iii) implies that X1 is P-convex for supports. With Grothendieck duality: rt

C∞ injective iff

∀ u ∈ R ˇ

P (Xc 1) :

• rt

C∞(ΦX1([u]∼)) = 0 ⇒ u = 0 outside a compact subset of X1

• Let K ⋐ X1, u ∈ R ˇ

P (Rd\K) with rt C∞(ΦX1([u]∼)) = 0, i.e.

∀ f ∈ C∞

P (X2) : 0 = rt C∞(ΦX1([u]∼)), fX2 = ˇ

P(∂)(ψu), f|X1X1 = ˇ P(∂)(ψu), fX2 = ΦX2([u]∼), fX2, i.e. ΦX2([u]∼) = 0 where ψ ∈ C∞(Rd), supp ψ ∩ K = ∅, supp (1 − ψ) ⊆ X1 compact. ΦX2([u]∼) = 0 ⇒ ∃ L ⋐ X2 : ψu ∈ E ′(L), ˇ P(∂)(ψu) ∈ E ′(X1)

iii)

⇒ ψu ∈ E ′(X1)

Thomas Kalmes PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 4]

Given P non-constant, X1 ⊆ X2 ⊆ Rd open, X2 P-convex for supports. Tfae. i) X1 is P-convex for supports and rD′ : D′

P (X2) → D′ P (X1), u → u|X1 has

dense range. ii) X1 is P-convex for supports and rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 has

dense range. iii) For every u ∈ E ′(X2) with supp ˇ P(∂)u ⊆ X1 it holds supp u ⊆ X1. iv) For every ϕ ∈ D(X2) with supp ˇ P(∂)ϕ ⊆ X1 it holds supp ϕ ⊆ X1. Proof of ”iii) ⇒ ii)”: iii) implies that X1 is P-convex for supports. With Grothendieck duality: rt

C∞ injective iff

∀ u ∈ R ˇ

P (Xc 1) :

• rt

C∞(ΦX1([u]∼)) = 0 ⇒ u = 0 outside a compact subset of X1

• Let K ⋐ X1, u ∈ R ˇ

P (Rd\K) with rt C∞(ΦX1([u]∼)) = 0, i.e.

∀ f ∈ C∞

P (X2) : 0 = rt C∞(ΦX1([u]∼)), fX2 = ˇ

P(∂)(ψu), f|X1X1 = ˇ P(∂)(ψu), fX2 = ΦX2([u]∼), fX2, i.e. ΦX2([u]∼) = 0 where ψ ∈ C∞(Rd), supp ψ ∩ K = ∅, supp (1 − ψ) ⊆ X1 compact. ΦX2([u]∼) = 0 ⇒ ∃ L ⋐ X2 : ψu ∈ E ′(L), ˇ P(∂)(ψu) ∈ E ′(X1)

iii)

⇒ ψu ∈ E ′(X1)

• Thomas Kalmes

PDO with a single characteristic direction 17 / 25

Theorem [7, Theorem 1]

Given P with ∃ N ∈ Rd, |N| = 1 : {ξ ∈ Rd; Pm(ξ) = 0} = span{N} and let X1 ⊆ X2 ⊆ Rd be open and P-convex for supports. Assume ∀ x ∈ Rd :

• C compact connected component of (Rd\X1) ∩ Hx ⇒ C X2
• ,

where Hx = x +

• span{N}

⊥. Then, both restriction maps rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 and rD′ : D′ P (X2) → D′ P (X1), u → u|X1

have dense range.

Thomas Kalmes PDO with a single characteristic direction 18 / 25

Theorem [7, Theorem 1]

Given P with ∃ N ∈ Rd, |N| = 1 : {ξ ∈ Rd; Pm(ξ) = 0} = span{N} and let X1 ⊆ X2 ⊆ Rd be open and P-convex for supports. Assume ∀ x ∈ Rd :

• C compact connected component of (Rd\X1) ∩ Hx ⇒ C X2
• ,

where Hx = x +

• span{N}

⊥. Then, both restriction maps rC∞ : C∞

P (X2) → C∞ P (X1), f → f|X1 and rD′ : D′ P (X2) → D′ P (X1), u → u|X1

have dense range.

Corollary [7, Corollary 3]

Let p ∈ N0, d ∈ N, d ≥ 2, and let Q ∈ C[X1, . . . , Xd−1] be elliptic of degree ≥ p + 1. Moreover, let Y1 ⊆ Y2 ⊆ Rd−1, I1 ⊆ I2 ⊆ R be open such that Y2 does not contain a compact connected component of Rd−1\Y1. Then, with P(∂) = ∂p

∂tp − Q(∂y) both restriction maps

rC∞ : C∞

P (Y2 × I2) → C∞ P (Y1 × I1) and rD′ : D′ P (Y2 × I2) → D′ P (Y1 × I1)

have dense range.

Thomas Kalmes PDO with a single characteristic direction 18 / 25

The linear topological invariant (Ω) for kernels

Thomas Kalmes PDO with a single characteristic direction 19 / 25

Let P(∂) : C∞(X) → C∞(X) be surjective. Given a locally convex space F, is P(∂) : C∞(X, F) → C∞(X, F) surjective?

Thomas Kalmes PDO with a single characteristic direction 20 / 25

Let P(∂) : C∞(X) → C∞(X) be surjective. Given a locally convex space F, is P(∂) : C∞(X, F) → C∞(X, F) surjective? ”Yes” if F is a Fr´ echet space (Grothendieck, 1955). In general ”No” for F = E′

b, the strong dual of a Fr´

echet space E (Vogt, 1983).

Thomas Kalmes PDO with a single characteristic direction 20 / 25

Let P(∂) : C∞(X) → C∞(X) be surjective. Given a locally convex space F, is P(∂) : C∞(X, F) → C∞(X, F) surjective? ”Yes” if F is a Fr´ echet space (Grothendieck, 1955). In general ”No” for F = E′

b, the strong dual of a Fr´

echet space E (Vogt, 1983). ”Yes” in case of F = s′ if and only if C∞

P (X) has (Ω) (Vogt, 1983).

”Yes” if C∞

P (X) has (Ω) and F = E′ b is the strong dual of a Fr´

echet space E with (DN) (Vogt, 1983).

Thomas Kalmes PDO with a single characteristic direction 20 / 25

Let P(∂) : C∞(X) → C∞(X) be surjective. Given a locally convex space F, is P(∂) : C∞(X, F) → C∞(X, F) surjective? ”Yes” if F is a Fr´ echet space (Grothendieck, 1955). In general ”No” for F = E′

b, the strong dual of a Fr´

echet space E (Vogt, 1983). ”Yes” in case of F = s′ if and only if C∞

P (X) has (Ω) (Vogt, 1983).

”Yes” if C∞

P (X) has (Ω) and F = E′ b is the strong dual of a Fr´

echet space E with (DN) (Vogt, 1983). C∞

P (X) has (Ω) if and only if

∃ T : s → C∞

P (X) linear, continuous, surjective (Vogt, Wagner 1980)

(Thus: C∞

P (X) has (Ω) ⇒ C∞ P (X) has a (absolute) Schauder basis)

Thomas Kalmes PDO with a single characteristic direction 20 / 25

E Fr´ echet space with a fundamental sequence of seminorms · 1 ≤ · 2 ≤ . . . (E.g. E = H (X), · k := · 0,Kk, for a compact exhaustion (Kk)k∈N of X). For u ∈ E′, k ∈ N, set u∗

k := supf∈E,fk≤1 |u, f|, dual seminorm to · k.

E has (Ω) :⇔ ∀ k ∈ N ∃ l ≥ k ∀ n ≥ l ∃λ ∈ (0, 1), C > 0 : · ∗

l ≤ C · ∗ λ k

· ∗ 1−λ

n

.

Thomas Kalmes PDO with a single characteristic direction 21 / 25

E Fr´ echet space with a fundamental sequence of seminorms · 1 ≤ · 2 ≤ . . . (E.g. E = H (X), · k := · 0,Kk, for a compact exhaustion (Kk)k∈N of X). For u ∈ E′, k ∈ N, set u∗

k := supf∈E,fk≤1 |u, f|, dual seminorm to · k.

E has (Ω) :⇔ ∀ k ∈ N ∃ l ≥ k ∀ n ≥ l ∃λ ∈ (0, 1), C > 0 : · ∗

l ≤ C · ∗ λ k

· ∗ 1−λ

n

. Example: P(ξ1, ξ2) = 1

2

• ∂1 + i∂2
• ⇒ C∞

P (B(0, 1)) = H (B(0, 1)), fundamental

sequence of seminorms fk := sup|z|≤1−

1 k+1 |f(z)|, k ∈ N.

Grothendieck-K¨

• the duality:

H (B(0, 1))′ ∼ = {u ∈ H (C\B(0, 1)); lim

|z|→∞ u(z) = 0}

as well as {u ∈ H (B(0, 1))′; u∗

k < ∞} ∼

= {u ∈ H (C\B(0, 1 − 1 k + 1)); lim

|z|→∞ u(z) = 0}

with u∗

k = sup|z|=1−1/(k+1) |u(z)|.

Thomas Kalmes PDO with a single characteristic direction 21 / 25

E Fr´ echet space with a fundamental sequence of seminorms · 1 ≤ · 2 ≤ . . . (E.g. E = H (X), · k := · 0,Kk, for a compact exhaustion (Kk)k∈N of X). For u ∈ E′, k ∈ N, set u∗

k := supf∈E,fk≤1 |u, f|, dual seminorm to · k.

E has (Ω) :⇔ ∀ k ∈ N ∃ l ≥ k ∀ n ≥ l ∃λ ∈ (0, 1), C > 0 : · ∗

l ≤ C · ∗ λ k

· ∗ 1−λ

n

. Example: P(ξ1, ξ2) = 1

2

• ∂1 + i∂2
• ⇒ C∞

P (B(0, 1)) = H (B(0, 1)), fundamental

sequence of seminorms fk := sup|z|≤1−

1 k+1 |f(z)|, k ∈ N.

Grothendieck-K¨

• the duality:

H (B(0, 1))′ ∼ = {u ∈ H (C\B(0, 1)); lim

|z|→∞ u(z) = 0}

as well as {u ∈ H (B(0, 1))′; u∗

k < ∞} ∼

= {u ∈ H (C\B(0, 1 − 1 k + 1)); lim

|z|→∞ u(z) = 0}

with u∗

k = sup|z|=1−1/(k+1) |u(z)|. Hadamard’s Three Circles Theorem applied

to z → ˜ u(z) := u( 1

z) (holomorphic in a neighborhood of B[0, 1 + 1 k]) gives

u∗

l =

sup

|z|=1+ 1

l

|˜ u(z)| ≤

• sup

|z|=1+ 1

k

|˜ u(z)| λ sup

|z|=1+ 1

n

|˜ u(z)| 1−λ = u∗ λ

k u∗ 1−λ n

for k ≤ l ≤ n with λ = ln(1+1/l)−ln(1+1/n)

ln(1+1/k)−ln(1+1/n), so H (B(0, 1)) has (Ω)

Thomas Kalmes PDO with a single characteristic direction 21 / 25

C∞

P (X) has (Ω) if

P is elliptic, X arbitrary (Vogt, 1983) P is hypoelliptic, X convex (Vogt, 1983) P is hypoelliptic, X ⊆ R2 P-convex for supports (K. 2012)

Thomas Kalmes PDO with a single characteristic direction 22 / 25

C∞

P (X) has (Ω) if

P is elliptic, X arbitrary (Vogt, 1983) P is hypoelliptic, X convex (Vogt, 1983) P is hypoelliptic, X ⊆ R2 P-convex for supports (K. 2012) However, for all d ≥ 3 there are hypoelliptic P and X ⊆ Rd P-convex for supports such that C∞

P (X) does not have (Ω). (K. 2012)

Thomas Kalmes PDO with a single characteristic direction 22 / 25

C∞

P (X) has (Ω) if

P is elliptic, X arbitrary (Vogt, 1983) P is hypoelliptic, X convex (Vogt, 1983) P is hypoelliptic, X ⊆ R2 P-convex for supports (K. 2012) However, for all d ≥ 3 there are hypoelliptic P and X ⊆ Rd P-convex for supports such that C∞

P (X) does not have (Ω). (K. 2012)

For α, m ∈ Nd

0 define |α : m| := d j=1 αj/mj; P is called semi-elliptic if it is

possible to write P(ξ) =

• |α:m|≤1

aαξα such that ∀ ξ ∈ Rd\{0} : P 0(iξ) :=

|α:m|=1 aαi|α|ξα = 0.

Thomas Kalmes PDO with a single characteristic direction 22 / 25

C∞

P (X) has (Ω) if

P is elliptic, X arbitrary (Vogt, 1983) P is hypoelliptic, X convex (Vogt, 1983) P is hypoelliptic, X ⊆ R2 P-convex for supports (K. 2012) However, for all d ≥ 3 there are hypoelliptic P and X ⊆ Rd P-convex for supports such that C∞

P (X) does not have (Ω). (K. 2012)

For α, m ∈ Nd

0 define |α : m| := d j=1 αj/mj; P is called semi-elliptic if it is

possible to write P(ξ) =

• |α:m|≤1

aαξα such that ∀ ξ ∈ Rd\{0} : P 0(iξ) :=

|α:m|=1 aαi|α|ξα = 0.

P elliptic ⇒ P semi-elliptic ⇒ P hypoelliptic Examples: P(ξ) = ξd − d−1

j=1 ξ2 j ; more general P(ξ) = ξp d − Q(ξ1, . . . , ξd−1),

with p ∈ N, p odd, Q ∈ C[X1, . . . , Xd−1] elliptic, deg(Q) = m ≥ p + 1, coefficients of Qm real.

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C∞

P (X) has (Ω) if

P is elliptic, X arbitrary (Vogt, 1983) P is hypoelliptic, X convex (Vogt, 1983) P is hypoelliptic, X ⊆ R2 P-convex for supports (K. 2012) However, for all d ≥ 3 there are hypoelliptic P and X ⊆ Rd P-convex for supports such that C∞

P (X) does not have (Ω). (K. 2012)

For α, m ∈ Nd

0 define |α : m| := d j=1 αj/mj; P is called semi-elliptic if it is

possible to write P(ξ) =

• |α:m|≤1

aαξα such that ∀ ξ ∈ Rd\{0} : P 0(iξ) :=

|α:m|=1 aαi|α|ξα = 0.

P elliptic ⇒ P semi-elliptic ⇒ P hypoelliptic Examples: P(ξ) = ξd − d−1

j=1 ξ2 j ; more general P(ξ) = ξp d − Q(ξ1, . . . , ξd−1),

with p ∈ N, p odd, Q ∈ C[X1, . . . , Xd−1] elliptic, deg(Q) = m ≥ p + 1, coefficients of Qm real. P semi-elliptic ⇒ {ξ ∈ Rd; Pm(ξ) = 0} is a subspace of Rd.

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Theorem [6, Theorem 18]

Let P be semi-elliptic with principal part Pm and let X ⊆ Rd be open. If dX satisfies the minimum principle in x + {ξ ∈ Rd; Pm(ξ) = 0}⊥ for every x ∈ Rd then X is P-convex for supports and C∞

P (X) has (Ω).

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Theorem [6, Theorem 18]

Let P be semi-elliptic with principal part Pm and let X ⊆ Rd be open. If dX satisfies the minimum principle in x + {ξ ∈ Rd; Pm(ξ) = 0}⊥ for every x ∈ Rd then X is P-convex for supports and C∞

P (X) has (Ω).

Corollary

Let P be semi-elliptic such that {ξ ∈ Rd; Pm(ξ) = 0} = span{N} with |N| = 1. Tfae i) X is P-convex for supports. ii) X is P-convex for supports and C∞

P (X) has (Ω).

iii) ∀ x ∈ Rd : dX satisfies the minimum principle in x +

• span{N}

⊥.

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References [1] A. Grothendieck, Sur les espaces de solutions d’une classe g´ en´ erale d’´ equations aux d´ eriv´ ees partielles, J. Analyse Math. 2 (1953), 243–280. [2] A. Grothendieck, Produits tensoriels topologiques et espaces nucl´ eaires,

• Mem. Amer. Math. Soc. 16, 1955.

[3] L. H¨

• rmander, On the existence and the regularity of solutions of linear

pseudo-differential equations, Enseignement Math. (2) 17 (1971), 99-163. [4] T. Kalmes, Some results on surjectivity of augmented differential

• perators, J. Math. Anal. Appl. 386 (2012), no. 1, 125–134.

[5] T. Kalmes, The augmented operator of a surjective partial differential

• perator with constant coefficients need not be surjective, Bull. London
• Math. Soc. 44 (2012), no. 3, 610–614.

[6] T. Kalmes, Surjectivity of differential operators and linear topological invariants for spaces of zero solutions, Rev. Mat. Compl. 32 (2019), no. 1, 37-55. [7] T. Kalmes, An approximation theorem of Runge type for certain non-elliptic partial differential operators, arXiv-preprint 1804.08099, 2018.

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[8] B. Malgrange, Existence et approximation des solutions des ´ equations aux d´ eriv´ ees partielles et des ´ equations de convolution, Ann. Inst. Fourier Grenoble, 6 (1955-1956), 271-355. [9] S. Nakane, P-Convexity with respect to differential operators which act on linear subspaces, Proc. Japan Acad. 55 Ser. A Math. Sci. (1979), no. 9, 343–347. [10] J. Persson, The wave operator and P-convexity, Boll. Un. Mat. Ital. B (5) 18 (1981), 591–604. [11] K. Tintarev, On the geometry of P-convex sets for operators of real principle type, Israel J. Math. 64 (1988), no. 2, 195–206. [12] K. Tintarev, Characterization of P-convexity for supports in terms of tangent curves, J. Math. Anal. Appl. 164 (1992), no. 2, 590–596. [13] D. Vogt, M.J. Wagner, Charakterisierung der Quotientenr¨ aume von s und eine Vermutung von Martineau, Studia Math. 67 (1980), no. 3, 225–240. [14] D. Vogt, On the Solvability of P(D)f = g for vector valued functions, RIMS Kokyoroku 508 (1983), 168–181.

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