[PPT] - The equality of the homogeneous and the Gabor wave front set Patrik PowerPoint Presentation

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The equality of the homogeneous and the Gabor wave front set

Patrik Wahlberg

Universit` a di Torino Joint work with Ren´ e Schulz, G¨

ttingen

XXXIII Convegno Nazionale di Analisi Armonica 17–20 giugno 2013 Alba

SLIDE 2

Plan of the talk

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18

SLIDE 3

Plan of the talk

A brief reveiw of the C ∞ wave front set WF(u) of u ∈ D′(Rd)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18

SLIDE 4

Plan of the talk

A brief reveiw of the C ∞ wave front set WF(u) of u ∈ D′(Rd) The Gabor (global) wave front set WFG(u) of u ∈ S ′(Rd) (H¨

rmander 1991)
P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18

SLIDE 5

Plan of the talk

A brief reveiw of the C ∞ wave front set WF(u) of u ∈ D′(Rd) The Gabor (global) wave front set WFG(u) of u ∈ S ′(Rd) (H¨

rmander 1991)

The homogeneous wave front set HWF(u) (Nakamura 2005)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18

SLIDE 6

Plan of the talk

A brief reveiw of the C ∞ wave front set WF(u) of u ∈ D′(Rd) The Gabor (global) wave front set WFG(u) of u ∈ S ′(Rd) (H¨

rmander 1991)

The homogeneous wave front set HWF(u) (Nakamura 2005) Main result: WFG(u) = HWF(u) for u ∈ S ′(Rd)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18

SLIDE 7

Plan of the talk

A brief reveiw of the C ∞ wave front set WF(u) of u ∈ D′(Rd) The Gabor (global) wave front set WFG(u) of u ∈ S ′(Rd) (H¨

rmander 1991)

The homogeneous wave front set HWF(u) (Nakamura 2005) Main result: WFG(u) = HWF(u) for u ∈ S ′(Rd) A tool for the proof: global semiclassical pseudo-differential calculus

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18

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The C ∞ wave front set

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18

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The C ∞ wave front set

Precursor by M. Sato 1969 (hyperfunctions)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18

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The C ∞ wave front set

Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨

rmander 1971 in “Fourier integral operators I”
P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18

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The C ∞ wave front set

Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨

rmander 1971 in “Fourier integral operators I”

u ∈ D′(Rd), x0 ∈ Rd, ξ0 ∈ Rd \ {0}.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18

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The C ∞ wave front set

Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨

rmander 1971 in “Fourier integral operators I”

u ∈ D′(Rd), x0 ∈ Rd, ξ0 ∈ Rd \ {0}. (x0, ξ0) / ∈ WF(u) if ∃ϕ ∈ C ∞

c (Rd), ∃Γ ⊆ Rd \ {0} conic, open, and

ξ0 ∈ Γ, such that ϕ(x0) = 0 and sup

ξ∈Γ

ξN|F(uϕ)(ξ)| < ∞ ∀N ≥ 0 where ξ = (1 + |ξ|2)1/2

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18

SLIDE 13

The C ∞ wave front set

Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨

rmander 1971 in “Fourier integral operators I”

u ∈ D′(Rd), x0 ∈ Rd, ξ0 ∈ Rd \ {0}. (x0, ξ0) / ∈ WF(u) if ∃ϕ ∈ C ∞

c (Rd), ∃Γ ⊆ Rd \ {0} conic, open, and

ξ0 ∈ Γ, such that ϕ(x0) = 0 and sup

ξ∈Γ

ξN|F(uϕ)(ξ)| < ∞ ∀N ≥ 0 where ξ = (1 + |ξ|2)1/2 x0 ξ0 Γ

Figure: x-space and the dual ξ-space

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18

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Basic properties of the C ∞ wave front set

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18

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Basic properties of the C ∞ wave front set

π(WF(u)) = sing supp(u), u ∈ D′(Rd), π(x, ξ) = x

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18

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Basic properties of the C ∞ wave front set

π(WF(u)) = sing supp(u), u ∈ D′(Rd), π(x, ξ) = x WF(u) = ∅ ⇐ ⇒ u ∈ C ∞

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18

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Basic properties of the C ∞ wave front set

π(WF(u)) = sing supp(u), u ∈ D′(Rd), π(x, ξ) = x WF(u) = ∅ ⇐ ⇒ u ∈ C ∞ H¨

rmander classes: m ∈ R, 0 ≤ δ < ρ ≤ 1

a ∈ Sm

ρ,δ

if |∂α

x ∂β ξ a(x, ξ)| ≤ Cα,βξm+δ|α|−ρ|β|,

x, ξ ∈ Rd

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18

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Basic properties of the C ∞ wave front set

π(WF(u)) = sing supp(u), u ∈ D′(Rd), π(x, ξ) = x WF(u) = ∅ ⇐ ⇒ u ∈ C ∞ H¨

rmander classes: m ∈ R, 0 ≤ δ < ρ ≤ 1

a ∈ Sm

ρ,δ

if |∂α

x ∂β ξ a(x, ξ)| ≤ Cα,βξm+δ|α|−ρ|β|,

x, ξ ∈ Rd Characteristic set of a ∈ Sm

ρ,δ: (x0, ξ0) ∈ (Rd × (Rd \ 0)) \ char(a) if

|a(x, ξ)| ≥ εξm, (x, ξ) ∈ Γ, |ξ| ≥ A ≥ 0, ε > 0, where (x0, ξ0) ∈ Γ, and Γ is open and conic in ξ: (x, ξ) ∈ Γ, t > 0 ⇒ (x, tξ) ∈ Γ

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18

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Basic properties of the C ∞ wave front set

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18

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Basic properties of the C ∞ wave front set

Microsupport of a ∈ Sm

ρ,δ: (x0, ξ0) ∈ (Rd × (Rd \ 0)) \ µ supp(a) if

|∂α

x ∂β ξ a(x, ξ)| ≤ Cα,β,Mξ−M,

M ≥ 0, (x, ξ) ∈ Γ where Γ is open, conic in ξ, and (x0, ξ0) ∈ Γ.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18

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Basic properties of the C ∞ wave front set

Microsupport of a ∈ Sm

ρ,δ: (x0, ξ0) ∈ (Rd × (Rd \ 0)) \ µ supp(a) if

|∂α

x ∂β ξ a(x, ξ)| ≤ Cα,β,Mξ−M,

M ≥ 0, (x, ξ) ∈ Γ where Γ is open, conic in ξ, and (x0, ξ0) ∈ Γ. Weyl quantization: a ∈ Sm

ρ,δ, f ∈ S (Rd)

aw(x, D)f (x) = (2π)−d

R2d eix−y,ξa

x + y 2 , ξ

f (y) dy dξ
P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18

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Basic properties of the C ∞ wave front set

Microsupport of a ∈ Sm

ρ,δ: (x0, ξ0) ∈ (Rd × (Rd \ 0)) \ µ supp(a) if

|∂α

x ∂β ξ a(x, ξ)| ≤ Cα,β,Mξ−M,

M ≥ 0, (x, ξ) ∈ Γ where Γ is open, conic in ξ, and (x0, ξ0) ∈ Γ. Weyl quantization: a ∈ Sm

ρ,δ, f ∈ S (Rd)

aw(x, D)f (x) = (2π)−d

R2d eix−y,ξa

x + y 2 , ξ

f (y) dy dξ

Theorem 1 (Inclusions)

If a ∈ Sm

ρ,δ and u ∈ S ′(Rd) then

WF(aw(x, D)u) ⊆ WF(u) ∩ µ supp(a) WF(u) ⊆ WF(aw(x, D)u) ∪ char(a)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18

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Basic properties of the C ∞ wave front set

Microlocal characterization: WF(u) =

a∈S0

1,0: aw(x,D)u∈C ∞

char(a)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 6 / 18

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The Gabor wave front set

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 7 / 18

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The Gabor wave front set

Introduced by H¨

rmander in Quadratic hyperbolic operators,

Microlocal Analysis and Applications, LNM vol. 1495, L. Cattabriga,

L. Rodino (Eds.), pp. 118–160, 1991
P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 7 / 18

SLIDE 26

The Gabor wave front set

Introduced by H¨

rmander in Quadratic hyperbolic operators,

Microlocal Analysis and Applications, LNM vol. 1495, L. Cattabriga,

L. Rodino (Eds.), pp. 118–160, 1991

Shubin symbol classes: m ∈ R, a ∈ G m if |∂α

x ∂β ξ a(x, ξ)| ≤ Cα,β(x, ξ)m−|α|−|β|,

(x, ξ) ∈ R2d

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 7 / 18

SLIDE 27

The Gabor wave front set

Introduced by H¨

rmander in Quadratic hyperbolic operators,

Microlocal Analysis and Applications, LNM vol. 1495, L. Cattabriga,

L. Rodino (Eds.), pp. 118–160, 1991

Shubin symbol classes: m ∈ R, a ∈ G m if |∂α

x ∂β ξ a(x, ξ)| ≤ Cα,β(x, ξ)m−|α|−|β|,

(x, ξ) ∈ R2d A noncharacteristic point for a ∈ G m is a point in the phase space z0 = (x0, ξ0) ∈ T ∗(Rd) \ {(0, 0)} such that |a(x, ξ)| ≥ ε(x, ξ)m, (x, ξ) ∈ Γ, |(x, ξ)| ≥ A where A, ε > 0 and Γ ⊆ T ∗(Rd) \ {(0, 0)} is an open conic set such that z0 ∈ Γ

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 7 / 18

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z0 x ξ Γ

Figure: A cone in the phase space (x, ξ)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 8 / 18

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For a ∈ G m, char(a) ⊆ T ∗(Rd) \ {(0, 0)} is the complement of the noncharacteristic points

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 9 / 18

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For a ∈ G m, char(a) ⊆ T ∗(Rd) \ {(0, 0)} is the complement of the noncharacteristic points

Definition 2

The Gabor wave front set WFG(u) of u ∈ S ′(Rd) is the set of all phase space points (x, ξ) ∈ T ∗(Rd) \ {(0, 0)} such that a ∈ G m, aw(x, D)u ∈ S ⇒ (x, ξ) ∈ char(a)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 9 / 18

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For a ∈ G m, char(a) ⊆ T ∗(Rd) \ {(0, 0)} is the complement of the noncharacteristic points

Definition 2

The Gabor wave front set WFG(u) of u ∈ S ′(Rd) is the set of all phase space points (x, ξ) ∈ T ∗(Rd) \ {(0, 0)} such that a ∈ G m, aw(x, D)u ∈ S ⇒ (x, ξ) ∈ char(a) WFG(u) =

a∈G ∞: aw(x,D)u∈S

char(a)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 9 / 18

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For a ∈ G m, char(a) ⊆ T ∗(Rd) \ {(0, 0)} is the complement of the noncharacteristic points

Definition 2

The Gabor wave front set WFG(u) of u ∈ S ′(Rd) is the set of all phase space points (x, ξ) ∈ T ∗(Rd) \ {(0, 0)} such that a ∈ G m, aw(x, D)u ∈ S ⇒ (x, ξ) ∈ char(a) WFG(u) =

a∈G ∞: aw(x,D)u∈S

char(a)

Theorem 3 (H¨

rmander 1991)

Let u ∈ S ′(Rd). Then WFG(u) = ∅ if and only if u ∈ S (Rd).

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 9 / 18

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Theorem 4 (H¨

rmander 1991)

If u ∈ S ′(Rd) and a ∈ G m then WFG(aw(x, D)u) ⊆ WFG(u)

conesupp(a)

WFG(u) ⊆ WFG(aw(x, D)u)

char(a)

The conic support conesupp(a) of a ∈ D′(Rd) is the set of all x ∈ Rd \ {0} such that any conic open set Γx containing x satisfies: supp(a) ∩ Γx is not compact in Rd.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 10 / 18

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Theorem 4 (H¨

rmander 1991)

If u ∈ S ′(Rd) and a ∈ G m then WFG(aw(x, D)u) ⊆ WFG(u)

conesupp(a)

WFG(u) ⊆ WFG(aw(x, D)u)

char(a)

The conic support conesupp(a) of a ∈ D′(Rd) is the set of all x ∈ Rd \ {0} such that any conic open set Γx containing x satisfies: supp(a) ∩ Γx is not compact in Rd. supp(a) conesupp(a)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 10 / 18

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Microlocal characterization

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 11 / 18

SLIDE 36

Microlocal characterization

Translation Txf (y) = f (y − x), modulation Mξf (y) = eiy,ξf (y)

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 11 / 18

SLIDE 37

Microlocal characterization

Translation Txf (y) = f (y − x), modulation Mξf (y) = eiy,ξf (y) ϕ ∈ S (Rd) \ {0} window function. The short-time Fourier transform (STFT) of u ∈ S ′(Rd) is Vϕu(x, ξ) = (u, MξTxϕ) = F(uTxϕ)(ξ), x, ξ ∈ Rd

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 11 / 18

SLIDE 38

Microlocal characterization

Translation Txf (y) = f (y − x), modulation Mξf (y) = eiy,ξf (y) ϕ ∈ S (Rd) \ {0} window function. The short-time Fourier transform (STFT) of u ∈ S ′(Rd) is Vϕu(x, ξ) = (u, MξTxϕ) = F(uTxϕ)(ξ), x, ξ ∈ Rd

Definition 5 (rapid decay of the STFT in a cone)

If u ∈ S ′(Rd) and ϕ ∈ S (Rd) \ {0} then for z0 ∈ T ∗(Rd) \ {(0, 0)} we say that z0 / ∈ WF ′(u) if there exists an open conic set Γz0 ⊆ T ∗(Rd) \ {(0, 0)} containing z0 such that sup

z∈Γz0

zN|Vϕu(z)| < ∞ ∀N ≥ 0.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 11 / 18

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Microlocal characterization

Translation Txf (y) = f (y − x), modulation Mξf (y) = eiy,ξf (y) ϕ ∈ S (Rd) \ {0} window function. The short-time Fourier transform (STFT) of u ∈ S ′(Rd) is Vϕu(x, ξ) = (u, MξTxϕ) = F(uTxϕ)(ξ), x, ξ ∈ Rd

Definition 5 (rapid decay of the STFT in a cone)

If u ∈ S ′(Rd) and ϕ ∈ S (Rd) \ {0} then for z0 ∈ T ∗(Rd) \ {(0, 0)} we say that z0 / ∈ WF ′(u) if there exists an open conic set Γz0 ⊆ T ∗(Rd) \ {(0, 0)} containing z0 such that sup

z∈Γz0

zN|Vϕu(z)| < ∞ ∀N ≥ 0.

Theorem 6 (microlocal characterization of WFG)

If u ∈ S ′(Rd) then WF ′(u) = WFG(u).

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 11 / 18

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The homogeneous wave front set

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 12 / 18

SLIDE 41

The homogeneous wave front set

Dilation by a small parameter: ah(z) = a(hz), 0 < h ≤ 1, z ∈ R2d.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 12 / 18

SLIDE 42

The homogeneous wave front set

Dilation by a small parameter: ah(z) = a(hz), 0 < h ≤ 1, z ∈ R2d.

Definition 7 (Nakamura 2005)

Let u ∈ S ′(Rd). A point (x, ξ) ∈ R2d \ {0} is not in the homogeneous wave front set HWF(u) if there exists a ∈ C ∞

c (R2d) with a(x, ξ) = 1

such that aw

h (x, D)uL2 = aw(hx, hD)uL2 = O(h∞) for h ∈ (0, 1].

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 12 / 18

SLIDE 43

The homogeneous wave front set

Dilation by a small parameter: ah(z) = a(hz), 0 < h ≤ 1, z ∈ R2d.

Definition 7 (Nakamura 2005)

Let u ∈ S ′(Rd). A point (x, ξ) ∈ R2d \ {0} is not in the homogeneous wave front set HWF(u) if there exists a ∈ C ∞

c (R2d) with a(x, ξ) = 1

such that aw

h (x, D)uL2 = aw(hx, hD)uL2 = O(h∞) for h ∈ (0, 1].

Here O(h∞) means O(hN) for any integer N ≥ 0.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 12 / 18

SLIDE 44

The homogeneous wave front set

Dilation by a small parameter: ah(z) = a(hz), 0 < h ≤ 1, z ∈ R2d.

Definition 7 (Nakamura 2005)

Let u ∈ S ′(Rd). A point (x, ξ) ∈ R2d \ {0} is not in the homogeneous wave front set HWF(u) if there exists a ∈ C ∞

c (R2d) with a(x, ξ) = 1

such that aw

h (x, D)uL2 = aw(hx, hD)uL2 = O(h∞) for h ∈ (0, 1].

Here O(h∞) means O(hN) for any integer N ≥ 0. Semiclassical analysis: ah(x, ξ) = a(x, hξ), x, ξ ∈ Rd, 0 < h ≤ 1.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 12 / 18

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The main result

Theorem 8 (Schulz, PW ’13)

If u ∈ S ′(Rd) then WFG(u) = HWF(u).

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 13 / 18

SLIDE 46

The main tool: a global semiclassical calculus of ΨDOs

Definition 9

Let a = a(z; h) ∈ C ∞(R2d × (0, 1]). Then a is a h-Shubin symbol of

rder m ∈ R, denoted a ∈ G m

h , if for all α ∈ N2d there exists a constant

Cα > 0 such that |∂α

z a(z; h)| ≤ Cαh|α|hzm−|α|,

z ∈ R2d, h ∈ (0, 1].

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 14 / 18

SLIDE 47

The main tool: a global semiclassical calculus of ΨDOs

Definition 9

Let a = a(z; h) ∈ C ∞(R2d × (0, 1]). Then a is a h-Shubin symbol of

rder m ∈ R, denoted a ∈ G m

h , if for all α ∈ N2d there exists a constant

Cα > 0 such that |∂α

z a(z; h)| ≤ Cαh|α|hzm−|α|,

z ∈ R2d, h ∈ (0, 1]. A typical case of an h-dependent symbol in G m

h is

ah(z) = a(hz), z ∈ R2d, h ∈ (0, 1], where a ∈ G m.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 14 / 18

SLIDE 48

Definition 10

Let aj ∈ G mj

h , j = 0, 1, . . . , where mj → −∞ as j → +∞, let

a ∈ C ∞(R2d × (0, 1]) and h ∈ (0, 1]. Set mj = maxk≥j mk. We write a ∼

∞

j=0

h

m0−mjaj

provided a −

N−1

j=0

h

m0−mjaj ∈ h m0− mNG mN h

, N ≥ 1, i.e. for any α ∈ N2d there exists CN,α > 0 such that

∂α

z

 a(z; h) −

N−1

j=0

h

m0−mjaj(z; h)

 

≤ CN,αh

m0− mN+|α|hz mN−|α|,

z ∈ R2d, h ∈ (0, 1].

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 15 / 18

SLIDE 49

Lemma 11

Suppose aj ∈ G mj

h

for j = 0, 1, . . . , where mj → −∞ as j → ∞, and suppose there exists ε > 0 such that supp(aj(·/h; h)) ∩ Bε = ∅, h ∈ (0, 1], j = 0, 1, . . . . Then there exists a ∈ G

m0 h

such that a ∼ ∞

0 h m0−mjaj.

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 16 / 18

SLIDE 50

Theorem 12

If u ∈ S ′(Rd) and a ∼ 0 then aw(x, D)uL2 = O(h∞).

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 17 / 18

SLIDE 51

Theorem 12

If u ∈ S ′(Rd) and a ∼ 0 then aw(x, D)uL2 = O(h∞). The Weyl product: (a#b)w(x, D) = aw(x, D)bw(x, D).

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 17 / 18

SLIDE 52

Theorem 12

If u ∈ S ′(Rd) and a ∼ 0 then aw(x, D)uL2 = O(h∞). The Weyl product: (a#b)w(x, D) = aw(x, D)bw(x, D).

Theorem 13

If a ∈ G m

h and b ∈ G n h then a#b ∈ G m+n h

and a#b(z) ∼

∞

j=0

h2j (iσ(D)/2)j j! (a ⊗ b)h−1

h(z,z).
P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 17 / 18

SLIDE 53

A consequence: an invariance result for the homogeneous wave front set

Theorem 14

Let u ∈ S ′(Rd) and suppose a ∈ C ∞

c (R2d), z0 ∈ R2d \ {0}, a(z0) = 1

and aw

h (x, D)uL2 = O(h∞),

h ∈ (0, 1]. Then there exists a relatively compact neighborhood U of z0, such that for any b ∈ C ∞

c (R2d) with supp(b) ⊆ U we have

bw

h (x, D)uL2 = O(h∞),

h ∈ (0, 1].

P. Wahlberg (Torino)

Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 18 / 18

SLIDE 54