The equality of the homogeneous and the Gabor wave front set Patrik Wahlberg Universit` a di Torino Joint work with Ren´ e Schulz, G¨ ottingen XXXIII Convegno Nazionale di Analisi Armonica 17–20 giugno 2013 Alba
Plan of the talk P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18
Plan of the talk A brief reveiw of the C ∞ wave front set WF ( u ) of u ∈ D ′ ( R d ) P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18
Plan of the talk A brief reveiw of the C ∞ wave front set WF ( u ) of u ∈ D ′ ( R d ) The Gabor (global) wave front set WF G ( u ) of u ∈ S ′ ( R d ) (H¨ ormander 1991) P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18
Plan of the talk A brief reveiw of the C ∞ wave front set WF ( u ) of u ∈ D ′ ( R d ) The Gabor (global) wave front set WF G ( u ) of u ∈ S ′ ( R d ) (H¨ ormander 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
Plan of the talk A brief reveiw of the C ∞ wave front set WF ( u ) of u ∈ D ′ ( R d ) The Gabor (global) wave front set WF G ( u ) of u ∈ S ′ ( R d ) (H¨ ormander 1991) The homogeneous wave front set HWF ( u ) (Nakamura 2005) Main result: WF G ( u ) = HWF ( u ) for u ∈ S ′ ( R d ) P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 2 / 18
Plan of the talk A brief reveiw of the C ∞ wave front set WF ( u ) of u ∈ D ′ ( R d ) The Gabor (global) wave front set WF G ( u ) of u ∈ S ′ ( R d ) (H¨ ormander 1991) The homogeneous wave front set HWF ( u ) (Nakamura 2005) Main result: WF G ( u ) = HWF ( u ) for u ∈ S ′ ( R d ) 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
The C ∞ wave front set P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18
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
The C ∞ wave front set Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨ ormander 1971 in “Fourier integral operators I” P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18
The C ∞ wave front set Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨ ormander 1971 in “Fourier integral operators I” u ∈ D ′ ( R d ) , x 0 ∈ R d , ξ 0 ∈ R d \ { 0 } . P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18
The C ∞ wave front set Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨ ormander 1971 in “Fourier integral operators I” u ∈ D ′ ( R d ) , x 0 ∈ R d , ξ 0 ∈ R d \ { 0 } . c ( R d ) , ∃ Γ ⊆ R d \ { 0 } conic, open, and ∈ WF ( u ) if ∃ ϕ ∈ C ∞ ( x 0 , ξ 0 ) / ξ 0 ∈ Γ , such that ϕ ( x 0 ) � = 0 and � ξ � N | F ( u ϕ )( ξ ) | < ∞ sup ∀ N ≥ 0 ξ ∈ Γ where � ξ � = ( 1 + | ξ | 2 ) 1 / 2 P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 3 / 18
The C ∞ wave front set Precursor by M. Sato 1969 (hyperfunctions) Introduced by H¨ ormander 1971 in “Fourier integral operators I” u ∈ D ′ ( R d ) , x 0 ∈ R d , ξ 0 ∈ R d \ { 0 } . c ( R d ) , ∃ Γ ⊆ R d \ { 0 } conic, open, and ∈ WF ( u ) if ∃ ϕ ∈ C ∞ ( x 0 , ξ 0 ) / ξ 0 ∈ Γ , such that ϕ ( x 0 ) � = 0 and � ξ � N | F ( u ϕ )( ξ ) | < ∞ sup ∀ N ≥ 0 ξ ∈ Γ where � ξ � = ( 1 + | ξ | 2 ) 1 / 2 x 0 ξ 0 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
Basic properties of the C ∞ wave front set P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18
Basic properties of the C ∞ wave front set u ∈ D ′ ( R d ) , π ( WF ( u )) = sing supp ( u ) , π ( x , ξ ) = x P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18
Basic properties of the C ∞ wave front set u ∈ D ′ ( R d ) , π ( WF ( u )) = sing supp ( u ) , π ( x , ξ ) = x u ∈ C ∞ WF ( u ) = ∅ ⇐ ⇒ P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18
Basic properties of the C ∞ wave front set u ∈ D ′ ( R d ) , π ( WF ( u )) = sing supp ( u ) , π ( x , ξ ) = x u ∈ C ∞ WF ( u ) = ∅ ⇐ ⇒ ormander classes : m ∈ R , 0 ≤ δ < ρ ≤ 1 H¨ | ∂ α x ∂ β ξ a ( x , ξ ) | ≤ C α,β � ξ � m + δ | α |− ρ | β | , a ∈ S m x , ξ ∈ R d if ρ,δ P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18
Basic properties of the C ∞ wave front set u ∈ D ′ ( R d ) , π ( WF ( u )) = sing supp ( u ) , π ( x , ξ ) = x u ∈ C ∞ WF ( u ) = ∅ ⇐ ⇒ ormander classes : m ∈ R , 0 ≤ δ < ρ ≤ 1 H¨ | ∂ α x ∂ β ξ a ( x , ξ ) | ≤ C α,β � ξ � m + δ | α |− ρ | β | , a ∈ S m x , ξ ∈ R d if ρ,δ ρ,δ : ( x 0 , ξ 0 ) ∈ ( R d × ( R d \ 0 )) \ char ( a ) if Characteristic set of a ∈ S m | a ( x , ξ ) | ≥ ε � ξ � m , ( x , ξ ) ∈ Γ , | ξ | ≥ A ≥ 0 , ε > 0 , where ( x 0 , ξ 0 ) ∈ Γ , and Γ is open and conic in ξ : ( x , ξ ) ∈ Γ , t > 0 ⇒ ( x , t ξ ) ∈ Γ P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 4 / 18
Basic properties of the C ∞ wave front set P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18
Basic properties of the C ∞ wave front set ρ,δ : ( x 0 , ξ 0 ) ∈ ( R d × ( R d \ 0 )) \ µ supp ( a ) if Microsupport of a ∈ S m x ∂ β | ∂ α ξ a ( x , ξ ) | ≤ C α,β, M � ξ � − M , M ≥ 0 , ( x , ξ ) ∈ Γ where Γ is open, conic in ξ , and ( x 0 , ξ 0 ) ∈ Γ . P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18
Basic properties of the C ∞ wave front set ρ,δ : ( x 0 , ξ 0 ) ∈ ( R d × ( R d \ 0 )) \ µ supp ( a ) if Microsupport of a ∈ S m x ∂ β | ∂ α ξ a ( x , ξ ) | ≤ C α,β, M � ξ � − M , M ≥ 0 , ( x , ξ ) ∈ Γ where Γ is open, conic in ξ , and ( x 0 , ξ 0 ) ∈ Γ . Weyl quantization: a ∈ S m ρ,δ , f ∈ S ( R d ) �� � x + y � a w ( x , D ) f ( x ) = ( 2 π ) − d R 2 d e i � x − y ,ξ � a , ξ f ( y ) dy d ξ 2 P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18
Basic properties of the C ∞ wave front set ρ,δ : ( x 0 , ξ 0 ) ∈ ( R d × ( R d \ 0 )) \ µ supp ( a ) if Microsupport of a ∈ S m x ∂ β | ∂ α ξ a ( x , ξ ) | ≤ C α,β, M � ξ � − M , M ≥ 0 , ( x , ξ ) ∈ Γ where Γ is open, conic in ξ , and ( x 0 , ξ 0 ) ∈ Γ . Weyl quantization: a ∈ S m ρ,δ , f ∈ S ( R d ) �� � x + y � a w ( x , D ) f ( x ) = ( 2 π ) − d R 2 d e i � x − y ,ξ � a , ξ f ( y ) dy d ξ 2 Theorem 1 (Inclusions) If a ∈ S m ρ,δ and u ∈ S ′ ( R d ) then WF ( a w ( x , D ) u ) ⊆ WF ( u ) ∩ µ supp ( a ) WF ( a w ( x , D ) u ) ∪ char ( a ) WF ( u ) ⊆ P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 5 / 18
Basic properties of the C ∞ wave front set Microlocal characterization: � WF ( u ) = char ( a ) a ∈ S 0 1 , 0 : a w ( x , D ) u ∈ C ∞ P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 6 / 18
The Gabor wave front set P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 7 / 18
The Gabor wave front set Introduced by H¨ ormander 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
The Gabor wave front set Introduced by H¨ ormander 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 x ∂ β | ∂ α ξ a ( x , ξ ) | ≤ C α,β � ( x , ξ ) � m −| α |−| β | , ( x , ξ ) ∈ R 2 d P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 7 / 18
The Gabor wave front set Introduced by H¨ ormander 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 x ∂ β | ∂ α ξ a ( x , ξ ) | ≤ C α,β � ( x , ξ ) � m −| α |−| β | , ( x , ξ ) ∈ R 2 d A noncharacteristic point for a ∈ G m is a point in the phase space z 0 = ( x 0 , ξ 0 ) ∈ T ∗ ( R d ) \ { ( 0 , 0 ) } such that | a ( x , ξ ) | ≥ ε � ( x , ξ ) � m , ( x , ξ ) ∈ Γ , | ( x , ξ ) | ≥ A where A , ε > 0 and Γ ⊆ T ∗ ( R d ) \ { ( 0 , 0 ) } is an open conic set such that z 0 ∈ Γ P. Wahlberg (Torino) Homogeneous and Gabor wave front set XXXIII Conv. Naz. A.A. 7 / 18
ξ Γ z 0 x 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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