Riesz transforms on group von Neumann algebras

Transcription

1 Riesz transforms on group von Neumann algebras Tao Mei Wayne State University Wuhan Joint work with M. Junge and J. Parcet June 17, 014

2 Classical Riesz transform f L (R n ), R = 1. with = ( x 1, x j, x n ), = x = j x j with R i = i 1 Rf = (R 1 f, R f, R n f ) the i th Riesz transform, R i f (ξ) = c ξ i ξ f (ξ), ξ R n. (Riesz; Stein/Meyer-Bakry/Pisier/Gundy/Varopoulos and many others) ( R j f ) 1 p f p, 1 < p <, j f p 1 f p, 1 < p <,

3 Carré du Champ P. A. Meyer s Gradient form = x on (R, dx); Chain rule: (f 1 f ) + ( f 1 )f + f 1 f = f 1 f. Given a generator of a Markov semigroup L (e.g. an elliptic operator), set Γ(f 1, f ) = L(f 1 f ) + L(f 1 )f + f 1 L(f ); R L (f ) = Γ(L 1 f, L 1 f )

4 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 = St 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.

5 Examples, P. A. Meyer s Gradient form L = : Laplace-Betrami operator Γ(f, f ) = f. (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. L = 1, S t = e tl on R n. Γ(f, f ) = 0 S t f + t S t f dt. G = F : the free group of two generaters. L : λ g g λ g, with g the word length of g F. Γ(λ g, λ h ) = g + h g 1 h λ g 1 h. P. A. Meyer s question When do we have Γ(f, f ) L p L 1 f Lp? ( )

6 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 0, then Γ(f, f ) L p L 1 f Lp holds for any 1 < p <. Γ(f, f ) 0 iff S t f S t f for S t = e tl. Γ 0 means Γ (f, f ) = LΓ(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 0 CD(0, ) criterion S t S t f S t f + S t f.)

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

8 Noncommutative extension Theorem (Junge;Junge/M noncommutative S t 010;) Assume L generates a noncommutative Markov semigroup on a semi finite von Neumann algebra M satisfying Γ 0, then L 1 f Lp(M) c p max{ Γ(f, f ) L p (M), Γ(f, f ) L p (M) } for any p <. f Lp(M) = (τ f p ) 1 p. Application to M. Rieffel s quantumn metric spaces;.. Question: Can we get an equivalence for all 1 < p <?

9 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) λ g generates a Markov semigroup on L (Ĝ). Γ(λ g, λ h ) = b(g) + b(h) b(g 1 h) = b(g), b(h) λ g 1 h λ g 1 h R L (f ) = Γ(L 1 f, L 1 f ) = k R k(f ) with R k : λ g b(g), v k b(g) λ g.

10 Semiclassical Riesz transform Examples G = R n (b, α, H) = (id, id, R n ). L = : e i ξ, ξ e i ξ,. R k = k 1. Let G = F : the free group generated by {h 1, h }. Λ = {δ g δ g ; g G} l (G) H = l (Λ). b : g δ g δ e H. b(g) = g, with g the word length of g F. S t : λ g e t g λ g. R L = k R k with R k : λ g b(g), v k λ g 1 g. Let v 1 = δ h1 δ e, v = δ h δ e, v 3 = δ h 1 1 δ e, v 4 = δ h 1 R 1 + R + R 3 + R 4 : λ g 1 λ g 1 g, g e δ e,

11 P. A. Meyer s question revisited P. A. Meyer s question G: (discrete) abelian group. ( R k (f ) ) 1 Lp(Ĝ) f Lp(Ĝ)? k R k (f )γ k Lp(Ω Ĝ) f Lp(Ĝ)? Fails for R k correspondence to 1. Recall G acts on H L (Ω). Revision of the question k R k (f ) γ k Lp(G L (Ω)) f Lp(Ĝ)? Yes! Junge/M/Parcet 014

12 L p Fourier multipliers on Discrete Groups G: (nonabelian) discrete group. e.g. G = Z, F, δ g, g G: the canonical basis of l (G). λ g : left regular representation of G on l (G), λ g (δ h ) = δ gh, for g, h G. L (Ĝ): the w closure of Span{λ g } s in B(l (G)). Example: G = Z, λ k = e ikθ, L (Ĝ) = L (T). τ: For f = g f g λ g, τf = f e. f L p (Ĝ) = [τ( f p )] 1 p, 1 p <. Example: G = Z, τf = ˆf (0) = f, L p (Ĝ) = L p (T). Question Find (nontrivial) Lp (Ĝ)(1 < p < ) bounded multipliers T m : λ g m(g)λ g.

13 L p bound of Riesz transforms via cocycles Theorem ( Junge/M/Parcet, 014) For any discrete group G with a cocycle (b, H, α), we have ( k R k f ) 1 Lp(Ĝ) + ( k R k f ) 1 Lp(Ĝ) f L p(ĝ), for any p. And f Lp(Ĝ) for any 1 < p <. ( inf ) 1 ( a j a j + ) 1 bj b j R j f =a j +b j Lp(Ĝ) j 1 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. j 1 Lp(Ĝ),

14 Classical Hörmander-Mihlin multipliers are averages of Riesz transforms Let G = R n. H = L (R n, dx x n+ε ). b : ξ R n e i ξ, 1 H. b(ξ) = ξ ε, L = ε : e iξx ξ ε e i ξ,x. For v H, let R v : e i ξ,x b(g), v b(ξ) ei ξ,x. Given T m : ˆf (ξ)e i ξ,x ˆf (ξ)m(ξ)e i ξ,x dξ, then T m = R v with v(x) = x n+ε m( ) ε.

15 Hörmander-Mihlin multipliers on a branch of free groups Given a branch B of G = F, let L p ( B) = {f = g B a g λ g ; f L p (Ĝ) = (τ f p ) 1 p < }. For m : Z + C, let T m f = g B m(g)a g λ g. Theorem ( Junge/M/Parcet, 014) Suppose m : Z + C satisfies then for any f L p ( B). sup m(j) + j m(j) m(j 1) < c j 1 T m f L p (Ĝ) c(p) c f L p (Ĝ)

16 Littlewood-Paley estimates. Theorem ( Junge/M /Parcet, 014) 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 < ( inf aj a j + b ) 1 j b j Λ j f =a j +b j L p (Ĝ) c(p) f L p (Ĝ), j 1 f L p (Ĝ) c(p) ( inf aj a j + b j bj Λ j f =a j +b j j 1 ) 1 L p (Ĝ).