Bilinear pseudodifferential operators of H ormander type Arp ad - PowerPoint PPT Presentation

Bilinear pseudodifferential operators of H¨ ormander type ´ Arp´ ad B´ enyi Department of Mathematics Western Washington University Bellingham, WA 98226 arpad.benyi@wwu.edu February Fourier Talks 2012

Outline of the talk Linear ψ DOs Some classical boundedness results Bilinear ψ DOs Results and comparison to linear case 2

Outline of the talk Linear ψ DOs Some classical boundedness results Bilinear ψ DOs Results and comparison to linear case 2

Outline of the talk Linear ψ DOs Some classical boundedness results Bilinear ψ DOs Results and comparison to linear case 2

Outline of the talk Linear ψ DOs Some classical boundedness results Bilinear ψ DOs Results and comparison to linear case 2

Fourier analysis For a function f , two complementary representations: The function f ( x ) itself (spatial behavior) The Fourier transform � f ( ξ ) (frequency behavior) � R d f ( x ) e − ix · ξ dx � f ( ξ ) = � f ( ξ ) e ix · ξ d ξ f ( x ) = (2 π ) − d � R d 3

Fourier analysis For a function f , two complementary representations: The function f ( x ) itself (spatial behavior) The Fourier transform � f ( ξ ) (frequency behavior) � R d f ( x ) e − ix · ξ dx � f ( ξ ) = � f ( ξ ) e ix · ξ d ξ f ( x ) = (2 π ) − d � R d 3

Linear multipliers The synthesis formula above is: � f ( ξ ) e ix · ξ d ξ R d (2 π ) − d � Id ( f )( x ) = � �� � m Translation invariant extension: � f ( ξ ) e ix · ξ d ξ R d m ( ξ ) � T m ( f )( x ) = Theorem (Mihlin, 1956) If | ∂ β m ( ξ ) | � (1 + | ξ | ) −| β | , then T σ : L p → L p , 1 < p < ∞ . 4

Linear multipliers The synthesis formula above is: � f ( ξ ) e ix · ξ d ξ R d (2 π ) − d � Id ( f )( x ) = � �� � m Translation invariant extension: � f ( ξ ) e ix · ξ d ξ R d m ( ξ ) � T m ( f )( x ) = Theorem (Mihlin, 1956) If | ∂ β m ( ξ ) | � (1 + | ξ | ) −| β | , then T σ : L p → L p , 1 < p < ∞ . 4

Linear pseudodifferential operators ( ψ DOs ) Non-translation invariant extension: � f ( ξ ) e ix · ξ d ξ R d σ ( x , ξ ) � T σ ( f )( x ) = Theorem (Ching, 1972; a question of Nirenberg) ξ σ ( x , ξ ) | � (1 + | ξ | ) −| β | , then T σ : L 2 �→ L 2 . If | ∂ β Boundeness requires also some a priori smoothness in x ! 5

Linear pseudodifferential operators ( ψ DOs ) Non-translation invariant extension: � f ( ξ ) e ix · ξ d ξ R d σ ( x , ξ ) � T σ ( f )( x ) = Theorem (Ching, 1972; a question of Nirenberg) ξ σ ( x , ξ ) | � (1 + | ξ | ) −| β | , then T σ : L 2 �→ L 2 . If | ∂ β Boundeness requires also some a priori smoothness in x ! 5

Linear pseudodifferential operators ( ψ DOs ) Non-translation invariant extension: � f ( ξ ) e ix · ξ d ξ R d σ ( x , ξ ) � T σ ( f )( x ) = Theorem (Ching, 1972; a question of Nirenberg) ξ σ ( x , ξ ) | � (1 + | ξ | ) −| β | , then T σ : L 2 �→ L 2 . If | ∂ β Boundeness requires also some a priori smoothness in x ! 5

(Linear) H¨ ormander classes of symbols Let m ∈ R and 0 ≤ ρ, δ ≤ 1. A symbol σ ( x , ξ ) belongs to the ormander class S m H¨ ρ,δ if | ∂ α x ∂ β ξ σ ( x , ξ ) | � (1 + | ξ | ) m + δ | α |− ρ | β | x ∂ β In particular: σ ∈ S 0 1 , 0 ⇔ | ∂ α ξ σ ( x , ξ ) | � (1 + | ξ | ) −| β | . Theorem (Coifman-Meyer, ’70s) 1 , 0 , then T σ : L p → L p , 1 < p < ∞ . If σ ∈ S 0 6

(Linear) H¨ ormander classes of symbols Let m ∈ R and 0 ≤ ρ, δ ≤ 1. A symbol σ ( x , ξ ) belongs to the ormander class S m H¨ ρ,δ if | ∂ α x ∂ β ξ σ ( x , ξ ) | � (1 + | ξ | ) m + δ | α |− ρ | β | x ∂ β In particular: σ ∈ S 0 1 , 0 ⇔ | ∂ α ξ σ ( x , ξ ) | � (1 + | ξ | ) −| β | . Theorem (Coifman-Meyer, ’70s) 1 , 0 , then T σ : L p → L p , 1 < p < ∞ . If σ ∈ S 0 6

(Linear) H¨ ormander classes of symbols Let m ∈ R and 0 ≤ ρ, δ ≤ 1. A symbol σ ( x , ξ ) belongs to the ormander class S m H¨ ρ,δ if | ∂ α x ∂ β ξ σ ( x , ξ ) | � (1 + | ξ | ) m + δ | α |− ρ | β | x ∂ β In particular: σ ∈ S 0 1 , 0 ⇔ | ∂ α ξ σ ( x , ξ ) | � (1 + | ξ | ) −| β | . Theorem (Coifman-Meyer, ’70s) 1 , 0 , then T σ : L p → L p , 1 < p < ∞ . If σ ∈ S 0 6

Connection to Calder´ on-Zygmund theory Note that: S 0 1 , 0 ⊂ S 0 1 ,δ ⊂ S 0 1 , 1 . Theorem The class S 0 1 , 1 is the largest one such that T σ has a Calder´ on-Zygmund kernel. That is, � T σ ( f )( x ) = K ( x , y ) f ( y ) dy , where K ( x , y ) satisfies | ∂ α x ∂ β y K ( x , y ) | � | x − y | − n −| α |−| β | . In particular, T σ : L p → L p ⇔ T σ : L 2 → L 2 . 1 ,δ : L 2 → L 2 , 0 ≤ δ < 1 but S 0 1 , 1 : L 2 �→ L 2 S 0 7

Connection to Calder´ on-Zygmund theory Note that: S 0 1 , 0 ⊂ S 0 1 ,δ ⊂ S 0 1 , 1 . Theorem The class S 0 1 , 1 is the largest one such that T σ has a Calder´ on-Zygmund kernel. That is, � T σ ( f )( x ) = K ( x , y ) f ( y ) dy , where K ( x , y ) satisfies | ∂ α x ∂ β y K ( x , y ) | � | x − y | − n −| α |−| β | . In particular, T σ : L p → L p ⇔ T σ : L 2 → L 2 . 1 ,δ : L 2 → L 2 , 0 ≤ δ < 1 but S 0 1 , 1 : L 2 �→ L 2 S 0 7

Connection to Calder´ on-Zygmund theory Note that: S 0 1 , 0 ⊂ S 0 1 ,δ ⊂ S 0 1 , 1 . Theorem The class S 0 1 , 1 is the largest one such that T σ has a Calder´ on-Zygmund kernel. That is, � T σ ( f )( x ) = K ( x , y ) f ( y ) dy , where K ( x , y ) satisfies | ∂ α x ∂ β y K ( x , y ) | � | x − y | − n −| α |−| β | . In particular, T σ : L p → L p ⇔ T σ : L 2 → L 2 . 1 ,δ : L 2 → L 2 , 0 ≤ δ < 1 but S 0 1 , 1 : L 2 �→ L 2 S 0 7

Connection to Calder´ on-Zygmund theory Note that: S 0 1 , 0 ⊂ S 0 1 ,δ ⊂ S 0 1 , 1 . Theorem The class S 0 1 , 1 is the largest one such that T σ has a Calder´ on-Zygmund kernel. That is, � T σ ( f )( x ) = K ( x , y ) f ( y ) dy , where K ( x , y ) satisfies | ∂ α x ∂ β y K ( x , y ) | � | x − y | − n −| α |−| β | . In particular, T σ : L p → L p ⇔ T σ : L 2 → L 2 . 1 ,δ : L 2 → L 2 , 0 ≤ δ < 1 but S 0 1 , 1 : L 2 �→ L 2 S 0 7

Some examples 1. Let a k ∈ C ∞ and | ∂ α x a k ( x ) | � 1. Define the PDO � a k ( x ) ∂ k T = x . | k |≤ m Then: T = T σ , where � a k ( x )( i ξ ) k . σ ( x , ξ ) = | k |≤ m We have: σ ∈ S m 1 , 0 . x a k ( x ) | � 2 k | α | and ψ ( ξ ) supported in 1 / 2 ≤ | ξ | ≤ 2. 2. Let | ∂ α Define ∞ � a k ( x ) ψ (2 − k ξ ) . σ ( x , ξ ) = k =1 We have: σ ∈ S 0 1 , 1 . 8

Some examples 1. Let a k ∈ C ∞ and | ∂ α x a k ( x ) | � 1. Define the PDO � a k ( x ) ∂ k T = x . | k |≤ m Then: T = T σ , where � a k ( x )( i ξ ) k . σ ( x , ξ ) = | k |≤ m We have: σ ∈ S m 1 , 0 . x a k ( x ) | � 2 k | α | and ψ ( ξ ) supported in 1 / 2 ≤ | ξ | ≤ 2. 2. Let | ∂ α Define ∞ � a k ( x ) ψ (2 − k ξ ) . σ ( x , ξ ) = k =1 We have: σ ∈ S 0 1 , 1 . 8

3. The heat operator n � ∂ 2 L = ∂ t − x 2 k k =1 has an approximate inverse T = T σ ( LT ∼ I ) and σ ∈ S − 1 1 / 2 , 0 . 9

The classes S 0 ρ,ρ Motivation Kumano-go, Nagase-Shinkai (’70s): applications to parabolic and semi-elliptic operators Theorem (Calder´ on-Vaillancourt, 1970) 0 , 0 , then T σ : L 2 → L 2 (but not on L p , p � = 2 , in general). If σ ∈ S 0 Recall that x ∂ β σ ∈ S 0 0 , 0 ⇔ | ∂ α ξ σ ( x , ξ ) | � 1 . Theorem (Cordes, 1975) ρ,ρ , 0 ≤ ρ < 1 , then T σ : L 2 → L 2 . If σ ∈ S 0 10

The classes S 0 ρ,ρ Motivation Kumano-go, Nagase-Shinkai (’70s): applications to parabolic and semi-elliptic operators Theorem (Calder´ on-Vaillancourt, 1970) 0 , 0 , then T σ : L 2 → L 2 (but not on L p , p � = 2 , in general). If σ ∈ S 0 Recall that x ∂ β σ ∈ S 0 0 , 0 ⇔ | ∂ α ξ σ ( x , ξ ) | � 1 . Theorem (Cordes, 1975) ρ,ρ , 0 ≤ ρ < 1 , then T σ : L 2 → L 2 . If σ ∈ S 0 10

The classes S m ρ, 0 Theorem (Fefferman-Stein, 1972) ρ, 0 , 0 < ρ < 1 , − (1 − ρ ) n / 2 < m ≤ 0 , then T σ : L 2 → L 2 . If σ ∈ S m Theorem (Fefferman, 1973) , 0 < ρ ≤ 1 , then T σ : L ∞ → BMO. If σ ∈ S − (1 − ρ ) n / 2 ρ, 0 Fefferman’s result uses the fact (due to H¨ ormader, ’70s) that ρ,δ : L 2 → L 2 , 0 < δ < ρ ≤ 1 . S 0 11

The classes S m ρ, 0 Theorem (Fefferman-Stein, 1972) ρ, 0 , 0 < ρ < 1 , − (1 − ρ ) n / 2 < m ≤ 0 , then T σ : L 2 → L 2 . If σ ∈ S m Theorem (Fefferman, 1973) , 0 < ρ ≤ 1 , then T σ : L ∞ → BMO. If σ ∈ S − (1 − ρ ) n / 2 ρ, 0 Fefferman’s result uses the fact (due to H¨ ormader, ’70s) that ρ,δ : L 2 → L 2 , 0 < δ < ρ ≤ 1 . S 0 11