Riesz transforms on group von Neumann algebras Tao Mei Wayne State - PowerPoint PPT Presentation

Riesz transforms on group von Neumann algebras Tao Mei Wayne State University Wuhan Joint work with M. Junge and J. Parcet June 17, 2014

Classical Riesz transform f ∈ L 2 ( R n ), R = ∂ △ − 1 2 . with � ) , △ = − ∂ 2 ∂ 2 ∂ = ( ∂ , · · ∂ , · · ∂ ∂ x 2 = ∂ x 2 ∂ x 1 ∂ x j ∂ x n j j Rf = ( R 1 f , R 2 f , · · · R n f ) with R i = ∂ i ∆ − 1 2 the i − th Riesz transform, R i f ( ξ ) = c ξ i � � f ( ξ ) , ξ ∈ R n . | ξ | (Riesz; Stein/Meyer-Bakry/Pisier/Gundy/Varopoulos and many others) � 1 | R j f | 2 ) 2 � p ≃ � f � p , 1 < p < ∞ , � ( j 1 2 f � p , 1 < p < ∞ , � ∂ f � p ≃ � ∆

Carr´ e du Champ—P. A. Meyer’s Gradient form ∆ = − ∂ 2 x on ( R , dx ); Chain rule: − ∆( f 1 f 2 ) + (∆ f 1 ) f 2 + f 1 ∆ f 2 = 2 ∂ f 1 ∂ f 2 . Given a generator of a Markov semigroup L (e.g. an elliptic operator), set − L ( f ∗ 1 f 2 ) + L ( f ∗ 1 ) f 2 + f ∗ 2Γ( f 1 , f 2 ) = 1 L ( f 2 ); Γ( L − 1 2 f , L − 1 | R L ( f ) | 2 2 f ) =

Semiclassical Riesz transform —Markov Semigroups of Operators ( M , µ ): Sigma finite measure space, ( S t ) t ≥ 0 : a semigroup of operators on L ∞ ( M ) We say ( S t ) t is Markov, if ◮ S t are contractions on L ∞ ( M , µ ). ◮ S t are symmetric i.e. � S t f , g � = � f , S t g � for f , g ∈ L 1 ( M ) ∩ L ∞ ( M ). ◮ S t (1) = 1 ◮ S t ( f ) → f in the w ∗ topology for f ∈ L ∞ ( M ) . Infinitesimal generator: L = − ∂ S t ∂ t | t =0 ; S t = e − tL . More general case: L ∞ ( M ) replaced by semi finite von Neuman algebras. Abstract theories by E. Stein, Cowling, Mcintoch...., Junge/Xu, Le Merdy-Junge-Xu.

Examples, P. A. Meyer’s Gradient form ◮ L = ∆: Laplace-Betrami operator Γ( f , f ) = |∇ f | 2 . ◮ ( M , dx ): complete Riemannian manifold. Lf ( x ) = � i , j a ij ( x ) ∂ i ∂ j f + � i g i ( x ) ∂ i f . Γ( f , f ) = � i , j a i , j ∂ i f ∂ j f . 1 2 , S t = e − tL on R n . ◮ L = ∆ � ∞ 0 |∇ S t f | 2 + | ∂ t S t f | 2 dt . Γ( f , f ) = ◮ G = F 2 : the free group of two generaters. L : λ g �→ | g | λ g , with | g | the word length of g ∈ F 2 . Γ( λ g , λ h ) = | g | + | h |−| g − 1 h | λ g − 1 h . 2 P. A. Meyer’s question When do we have 1 � Γ( f , f ) � 2 2 f � L p ? 2 ≃ � L ( ∗ ) L p

Semiclassical Riesz transform—P.A. Meyer’s question Theorem (D.Bakry for diffusion S t 1986;) Assume L generates a diffusion Markov semigroup (on commutative L p spaces) satisfying Γ 2 ≥ 0, then 1 � Γ( f , f ) � 2 2 f � L p 2 ≃ � L L p holds for any 1 < p < ∞ . Γ( f , f ) ≥ 0 iff | S t f | 2 ≤ S t | f | 2 for S t = e − tL . Γ 2 ≥ 0 means Γ 2 ( f , f ) = − L Γ( f , f ) + Γ( Lf , f ) + Γ( f , Lf ) ≥ 0 . P. A. Meyer, D. Bakry, M. Emery, X. D. Li, F. Baudoin-N. Garofalo, etc. (Γ 2 ≥ 0 ⇔ CD (0 , ∞ ) criterion ⇔ 2 S t | S t f | 2 ≤ S 2 t | f | 2 + | S 2 t f | 2 .)

Fractional power of ∆ 1 2 on R n . P. A. Meyer’s inequality (*) Fails for L = ∆ 2 n n +1 , and any Schwarz function f on R n , For p ≤ � Γ( f , f ) � L 2 = ∞ p while 1 2 f � L p < ∞ . � L

Noncommutative extension Theorem (Junge;Junge/M noncommutative S t 2010;) Assume L generates a noncommutative Markov semigroup on a semi finite von Neumann algebra M satisfying Γ 2 ≥ 0, then 1 2 f � L p ( M ) ≤ c p max {� Γ( f , f ) � 2 2 ( M ) , � Γ( f ∗ , f ∗ ) � 2 � L 2 ( M ) } L p L p for any 2 ≤ p < ∞ . 1 � f � L p ( M ) = ( τ | f | p ) p . Application to M. Rieffel’s quantumn metric spaces;.. Question: Can we get an equivalence for all 1 < p < ∞ ?

Riesz transforms via cocycles G : discrete (abelian) group ( b , α, H ): cocycle of group actions on Hilbert space H , i.e. b : G �→ H , α : G �→ Aut ( H ) , α g b ( g − 1 h ) = b ( h ) − b ( g ) v k : orthonormal basis of H . L : λ g �→ � b ( g ) � 2 λ g generates a Markov semigroup on L ∞ ( � G ). � b ( g ) � 2 + � b ( h ) � 2 − � b ( g − 1 h ) � 2 Γ( λ g , λ h ) = λ g − 1 h 2 = � b ( g ) , b ( h ) � λ g − 1 h 2 f ) = � | R L ( f ) | 2 = Γ( L − 1 2 f , L − 1 k | R k ( f ) | 2 with R k : λ g → � b ( g ) , v k � � b ( g ) � λ g .

Semiclassical Riesz transform—Examples ◮ G = R n ( b , α, H ) = ( id , id , R n ). L = ∆: e i � ξ, ·� → −| ξ | 2 e i � ξ, ·� . R k = ∂ k ∆ − 1 2 . ◮ Let G = F 2 : the free group generated by { h 1 , h 2 } . Λ = { δ g − δ g − ; g ∈ G } ⊂ ℓ 2 ( G ) H = ℓ 2 (Λ). b : g → δ g − δ e ∈ H . � b ( g ) � 2 = | g | , with | g | the word length of g ∈ F 2 . S t : λ g → e − t | g | λ g . | R L · | 2 = � k | R k · | 2 with R k : λ g → � b ( g ) , v k � λ g . 1 | g | 2 Let v 1 = δ h 1 − δ e , v 2 = δ h 2 − δ e , v 3 = δ h − 1 1 − δ e , v 4 = δ h − 1 2 − δ e , 1 R 1 + R 2 + R 3 + R 4 : λ g → λ g , g � = e 1 | g | 2

P. A. Meyer’s question revisited P. A. Meyer’s question G : (discrete) abelian group. � 1 | R k ( f ) | 2 ) 2 � L p (ˆ � ( ≃ � f � L p (ˆ G ) ? G ) � � R k ( f ) γ k � L p (Ω × ˆ ≃ � f � L p (ˆ G ) ? G ) k 1 2 . Fails for R k correspondence to ∆ Recall G acts on H ≃ L 2 (Ω). Revision of the question � � R k ( f ) ⋊ γ k � L p ( G ⋊ L ∞ (Ω)) ≃ � f � L p (ˆ G ) ? k Yes! Junge/M/Parcet 2014

L p Fourier multipliers on Discrete Groups e.g. G = Z , F 2 , G : (nonabelian) discrete group. δ g , g ∈ G : the canonical basis of ℓ 2 ( G ). λ g : left regular representation of G on ℓ 2 ( G ), λ g ( δ h ) = δ gh , for g , h ∈ G . G ): the w ∗ closure of Span { λ g } ’s in B ( ℓ 2 ( G )). L ∞ (ˆ Example: G = Z , λ k = e ik θ , L ∞ (ˆ G ) = L ∞ ( T ). τ : For f = � g f g λ g , τ f = f e . 1 G ) = [ τ ( | f | p )] p , 1 ≤ p < ∞ . || f || L p ( � � Example: G = Z , τ f = ˆ f , L p (ˆ f (0) = G ) = L p ( T ). Question Find (nontrivial) L p ( � G )(1 < p < ∞ ) bounded multipliers T m : λ g → m ( g ) λ g .

L p bound of Riesz transforms via cocycles Theorem ( Junge/M/Parcet, 2014) For any discrete group G with a cocycle ( b , H , α ), we have � � 1 1 | R k f | 2 ) | R k f ∗ | 2 ) 2 � L p ( � 2 � L p ( � � ( G ) + � ( G ) ≃ � f � L p ( � G ) , k k for any p ≥ 2. And � � � � 1 2 � � � � � 1 2 � � � � � � b j � a ∗ b ∗ � f � L p ( � G ) ≃ inf G ) + � j a j � � � G ) , j L p ( � L p ( � R j f = a j + b j j ≥ 1 j ≥ 1 for any 1 < p < 2. The equivalence constant only depends on p . Pisier’s method+Khintchine inequality for crossed products. � b j is a twist of b j coming from the Khintchine inequality for crossed products.

Classical H¨ ormander-Mihlin multipliers are averages of Riesz transforms ◮ Let G = R n . | x | n +2 ε ). b : ξ ∈ R n → e i � ξ, ·� − 1 ∈ H . dx H = L 2 ( R n , � b ( ξ ) � 2 = | ξ | 2 ε , L = − ∆ ε : e i ξ x → −| ξ | 2 ε e i � ξ, x � . ◮ For v ∈ H , let R v : e i � ξ, x � → � b ( g ) , v � � b ( ξ ) � e i � ξ, x � . � ˆ � ˆ f ( ξ ) e i � ξ, x � → f ( ξ ) m ( ξ ) e i � ξ, x � d ξ , then Given T m : T m = R v with v ( x ) = | x | n +2 ε � m ( · ) | · | ε .

H¨ ormander-Mihlin multipliers on a branch of free groups Given a branch B of G = F ∞ , let � 1 L p ( � G ) = ( τ | f | p ) p < ∞} . B ) = { f = a g λ g ; � f � L p ( � g ∈ B For m : Z + → C , let � T m f = m ( g ) a g λ g . g ∈ B Theorem ( Junge/M/Parcet, 2014) Suppose m : Z + → C satisfies sup | m ( j ) | + j | m ( j ) − m ( j − 1) | < c j ≥ 1 then � T m f � L p ( � G ) � c ( p ) c � f � L p ( � G ) for any f ∈ L p ( � B ) .

Littlewood-Paley estimates. Theorem ( Junge/M /Parcet, 2014) Consider a standard Littlewood-Paley partition of unity ( ϕ j ) j ≥ 1 in R + . Let � Λ j : λ ( g ) �→ ϕ j ( | g | ) λ ( g ) denote the corresponding radial multipliers in L ( F ∞ ). Then, the following estimates hold for f ∈ L p ( � B ) and 1 < p < 2 � 2 � � � � 1 � � j a j + � b j � a ∗ b ∗ inf � � G ) � c ( p ) � f � L p ( � G ) , j L p ( � Λ j f = a j + b j j ≥ 1 � � � � 1 2 � � � a ∗ j a j + b j b ∗ � f � L p ( � G ) � c ( p ) inf � � G ) . j L p ( � Λ j f = a j + b j j ≥ 1