Pisier s inequality revisited

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1 Pisier s inequality revisited Tuomas Hytönen Department of Mathematics and Statistics University of Helsinki POB 68, FI-14 Helsinki, Finland tuomashytonen@helsinkifi Assaf Naor Courant Institute of Mathematical Sciences New York University 251 Mercer Street, New York NY, 112, USA naor@cimsnyuedu Abstract Given a Banach space X, for n N and p (1, ) we investigate the smallest constant P (, ) for which every n-tuple of functions f 1,, f n : { 1, 1} n X satisfies p j f j (ε) dµ(ε) P p p δ j f j (ε) dµ(ε)dµ(δ), where µ is the uniform probability measure on the discrete hypercube { 1, 1} n and { j } n and n j are the hypercube partial derivatives and the hypercube Laplacian, respectively Denoting this constant by P n p (X), we show that P n p (X) for every Banach space (X, ) This extends the classical Pisier inequality, which corresponds to the special case f j 1 j f for some f : { 1, 1} n X We show that sup n N P n p (X) < if either the dual X is a UMD + Banach space, or for some θ (, 1) we have X [H, Y ] θ, where H is a Hilbert space and Y is an arbitrary Banach space It follows that sup n N P n p (X) < if X is a Banach lattice of nontrivial type k1 1 k 1

2 2 T Hytönen and A Naor 1 Introduction Fix a Banach space (X, ) and n N For every f : { 1, 1} n X and j {1,, n} the hypercube jth partial derivative of f, which is denoted j f : { 1, 1} n X, is defined as (11) j f(ε) def f(ε) f(ε 1,, ε j 1, ε j, ε j+1,, ε n ) 2 The hypercube Laplacian of f, denoted f : { 1, 1} n X, is (12) f(ε) def j f(ε) It is immediate to check that is invertible on the space of all mean zero functions f : { 1, 1} n X Below 1 is understood to be defined for every f : { 1, 1} n X by setting 1 f 1 f Here f f f(δ)dµ(δ), where µ denotes the uniform probability measure on { 1, 1} n The following inequality is due to Pisier [28] Throughout this paper the asymptotic notation, indicates the corresponding inequalities up to universal constant factors We will also denote equivalence up to universal constant factors by, ie, A B is the same as (A B) (A B) Theorem 11 (Pisier s inequality) For every Banach space (X, ), every n N, every p [1, ] and every f : { 1, 1} n X, we have (13) ( p ) 1/p f(ε) f(δ)dµ(δ) dµ(ε) { 1,1} n ( p 1/p log n δ j j f(ε) dµ(ε)dµ(δ)) Due to the application of Pisier s inequality to the theory of nonlinear type (see [28, 25, 12, 23]), it is of great interest to understand when (13) holds true with the log n term replaced by a constant that may depend on the geometry of X but is independent of n Talagrand proved [3] that the log n term in (13) is asymptotically optimal for general Banach spaces X, Wagner proved [31] that the log n term in (13) can be replaced by a universal constant if p and X is a general Banach space, and in [25] it is shown that the log n term in (13) can be replaced by a constant that is independent of n if X is a UMD Banach space It remains an intriguing open question whether every Banach space of nontrivial type satisfies (13) with

3 Pisier s inequality revisited 3 the log n term replaced by a constant that is independent of n If true, this would resolve a 1976 question of Enflo [9] by establishing that Rademacher type p and Enflo type p coincide (see [25, 23] and Section 6 below) Here we obtain a new class of Banach spaces that satisfies a dimensionindependent Pisier inequality Our starting point is the following extension of Pisier s inequality Definition 12 (Pisier constant of X) The n-dimensional Pisier constant of X (with exponent p), denoted P n p(x), is the infimum over those P (, ) such that every f 1,, f n : { 1, 1} n X satisfy (14) ( p j f j (ε) dµ(ε) ( P ) 1/p p 1/p δ j f j (ε) dµ(ε)dµ(δ)) We also set P p (X) def sup P n p(x) n N Inequality (14) reduces to Pisier s inequality if we choose f j 1 j f for some f : { 1, 1} n X The generalized inequality (14) has the advantage of being well-behaved under duality, as explained in Section 2 The following theorem yields a logarithmic bound on P p n(x), thus extending Pisier s inequality Theorem 13 For every Banach space X, every p [1, ] and every n N, P n 1 p(x) k Our approach yields a quantitative improvement over Pisier s inequality only in lower order terms: an optimization of Pisier s argument (as carried out in [23]) shows that the O(log n) term in (13) can be taken to be at most log n + O(log log n), while Theorem 13 shows that this term can be taken to be log n + O(1) In [25] it was shown that the logarithmic term in (13) can be replaced by a constant that is independent of n if X is a UMD Banach space Recall that X is a UMD Banach space if for every p (1, ) there exists a constant β (, ) such that if {M j } n j is a p-integrable X-valued martingale k1

4 4 T Hytönen and A Naor defined on some probability space (Ω, P), then for every ε 1,, ε n { 1, 1} we have (15) Ω M + ε j (M j M j 1 ) pdp β p M n p dp Ω The infimum over those β (, ) for which (15) holds true is denoted β p (X) It can be shown (see [7]) that β p (X) p2 β p 1 2(X), so in order to define the UMD property it suffices to require the validity of (15) for p 2 UMD Banach spaces are known to be superreflexive [2, 1], and one also has β q (X ) β p (X), where q p/(p 1) (see eg [7]) In [1] Garling investigated the natural weakening of (15) in which the desired inequality is required to hold true in expectation over ε 1,, ε n { 1, 1} rather than for every ε 1,, ε n { 1, 1} Specifically, say that X is a UMD + Banach space if for every p (1, ) there exists a constant β (, ) such that if {M j } n j is a p-integrable X-valued martingale defined on some probability space (Ω, P) then (16) Ω M + ε j (M j M j 1 ) pdpdµ(ε) β p M n p dp Ω The infimum over those β for which (16) holds true is denoted β + p (X) Theorem 14 If X is a Banach space such that X is UMD + then the following inequality holds true Fix p (1, ) and n N For every function F : { 1, 1} n { 1, 1} n X and j {1,, n} denote F j (ε) def δ j F (ε, δ)dµ(δ) Then (17) ( p ) 1/p 1 j F j (ε) dµ(ε) ( ) 1/p β q + (X ) F (ε, δ) p dµ(ε)dµ(δ), where q p/(p 1) For every f 1,, f n : { 1, 1} n X, an application of Theorem 14 to the function F (ε, δ) n δ jf j (ε) yields the following estimate on the Pisier constant of a UMD + Banach space

5 Pisier s inequality revisited 5 Corollary 15 P p (X) β + q (X ) It is unknown if a UMD + Banach space must also be a UMD Banach space, though it seems reasonable to conjecture that there are UMD + spaces that are not UMD Regardless of this, Theorem 14 and Corollary 15 are conceptually different from the result of [25], which relies on the full force of the UMD condition, ie it requires the validity of (15) for every choice of signs ε 1,, ε n, while our argument needs such estimates to hold true only for an average choice of signs We also have a quantitative improvement: in [25] it was shown that Pisier s inequality holds true with the O(log n) term in (13) replaced by β p (X) β q (X ), while we obtain the same estimate with the O(log n) term in (13) replaced by β + q (X ) β q (X ) Geiss proved [11] that for every η (, 1) there is C η (, ) such that for every M > 1 there is a Banach space X that satisfies > β q (X ) C η β + q (X ) 2 η M Remark 11 Inequality (17) is an extension of the generalized Pisier inequality (14), but for general Banach spaces it behaves very differently: unlike the logarithmic behavior of Theorem 13, the best constant appearing in the right hand side of (17) for a general Banach space X must be at least a constant multiple of n, as exhibited by the case X L 1 (({ 1, 1} n, µ), R) and F : { 1, 1} n { 1, 1} n X given by F (ε, δ)(η) n i1 (1 + ε iδ i η i ) Suppose that θ (, 1) and X [H, Y ] θ, where H is a Hilbert space and Y is an arbitrary Banach space Here [, ] θ denotes complex interpolation (see [3]) Theorem 16 below shows that in this case P p (X) <, and therefore Pisier s inequality holds true with the log n term in (13) replaced by a constant that is independent of n Pisier proved [27] that every Banach lattice of nontrivial type (see [19]) is of the form [H, Y ] θ for some θ (, 1), so we thus obtain the desired dimension independence in Pisier s inequality for Banach lattices of nontrivial type This result does not follow from previously known cases in which a dimension-independent Pisier inequality has been proved, since, as shown by Bourgain [4, 5], there exist Banach lattices of nontrivial type which are not UMD Note, however, that we are still far from proving the conjectured dimension-independent Pisier inequality for Banach spaces with nontrivial type: any space of the form [H, Y ] θ admits an equivalent norm whose modulus of smoothness has power type 2/(1 + θ) (see [27, 8]), while there exist Banach spaces with nontrivial type that do not admit such an equivalent norm (see [14, 16, 15, 29])

6 6 T Hytönen and A Naor Theorem 16 Let X, Y be Banach spaces and let H be a Hilbert space Suppose that for some θ (, 1) we have X [H, Y ] θ Then for every p (1, ), 2 max{p, p/(p 1)} P p (X) 1 θ Remark 12 If r (2, ) then the O(log n) term in Pisier s inequality (13), when p 2 and X l r, can be replaced by O(r), due to the fact that β 2 + (l r ) r (this follows from Hitczenko s work [13], as explained to us by Mark Veraar) This bound also follows from Theorem 16 At the same time, an inspection of Talagrand s example in [3] shows that this term must be at least a constant multiple of log r Determining the correct order of magnitude as r of the constant in Pisier s inequality when X l r remains an interesting open problem 2 Duality The dimension n N will be fixed from now on For p [1, ] and a Banach space X, let L p (X) denote the vector-valued Lebesgue space L p (({ 1, 1} n, µ), X) Thus L p (L p (X)) can be naturally identified with the space L p (({ 1, 1} n { 1, 1} n, µ µ), X) For f L p (X) we denote its Fourier expansion by f f(a)w A, where the Walsh function W A : { 1, 1} n { 1, 1} corresponding to A {1,, n} is given by W A (ε 1,, ε n ) i A ε i, and the Fourier coefficient f(a) X is given by f(a) f(x)w A (x)dµ(x) Using this (standard) notation, we have i {1,, n}, f L p (X), i f f(a)w A, and f L p (X), f f L p (X), 1 f def A i A A f(a)w A, 1 A f(a)w A The Rademacher projection of f L p (X) is defined as usual by Rad(f) def f({i})w {i} i1

7 Pisier s inequality revisited 7 def We denote below Rad X Rad(L p (X)) and Rad def X (I Rad)(L p (X)) The dual of (Rad X, Lp(X)) is naturally identified with the quotient L q (X )/Rad X, where q p/(p 1) Define an operator S : L p (L p (X)) L p (X) by (21) F L p (L p (X)), S(F ) def 1 j F ({j}) Using this notation, Theorem 14 is nothing more than the following operator norm bound S Lp(L p(x)) L p(x) β + q (X ) The adjoint operator S : L q (X ) L q (L q (X )) is given by g L q (X ), δ { 1, 1} n, S (g)(δ) δ j 1 j g Therefore Theorem 14 has the following equivalent dual formulation Theorem 21 (Dual formulation of Theorem 14) Let Z be a UMD + Banach space Then for every q (1, ) and every g L q (Z) we have q 1/q δ j 1 j g dµ(δ) β q + (Z) g Lq(Z) L q(z) Theorem 21, and consequently also Theorem 14, will be proven in Section 3 Let T be the restriction of S to Rad Lp(X) Thus P n p(x) T RadLp(X) L p(x) T Lq(X ) L q(l q(x ))/Rad Lq(X ) The adjoint T : L q (X ) L q (L q (X ))/Rad L q(x ) is given by g L q (X ), δ { 1, 1} n, T (g) δ j 1 j g + Rad L q(x ) Therefore Theorem 13 has the following equivalent dual formulation Theorem 22 (Dual formulation of Theorem 13) Let Z be a Banach space and q [1, ] Then for every g L q (Z) we have inf Φ Rad Lq(Z) Φ(δ) + δ j 1 j g q L q(z) dµ(δ) 1/q ( k1 ) 1 g Lq(Z) k

8 8 T Hytönen and A Naor Theorem 22, and consequently also Theorem 13, will be proven in Section 3 Since [H, Y ] θ [H, Y ] θ (see [3]), we also have the following equivalent dual formulation of Theorem 16 Theorem 23 (Dual formulation of Theorem 16) Let H be a Hilbert space, W a Banach space, and θ (, 1) Set Z [H, W ] θ Then for every q (1, ) and g L q (Z), q 1/q inf Ψ(δ) + δ j 1 j g dµ(δ) Ψ Rad Lq(Z) L q(z) 2 max{q, q/(q 1)} g Lq(Z) 1 θ Theorem 23, and consequently also Theorem 16, will be proven in Section 5 3 Proof of Theorem 21 Fix q (1, ) and g L q (Z) Let S n denote the symmetric group on {1,, n} For σ S n and k {,, n} define gk σ L q(z) by (31) gk σ (ε) def ĝ(a)w A (ε) A {σ 1 (1),,σ 1 (k)} 1 g 2 n k δ σ 1 (k+1),,δ σ 1 (n) { 1,1} ( k ε σ 1 (i)e σ 1 (i) + i1 ik+1 δ σ 1 (i)e σ 1 (i) where here, and in what follows, e 1,, e n denotes the standard basis of R n Then {gk σ}n k is a Z-valued martingale with gσ n g and g σ ĝ( ), implying that (32) ( ) q 1/q δ k g σ k gk 1 σ dµ(δ) β q + (Z) g Lq(Z) k1 L q(z) In (32) we may replace {δ k } n k1 by {δ σ 1 (k)} n k1, since these two sequences of signs have the same joint distribution Then we make the change of variable j σ 1 (k), so that k σ(j) Averaging the resulting inequality over σ S n, and using the convexity of the norm, we see that (33) 1 ( ) q 1/q δ j g σ n! σ(j) gσ(j) 1 σ dµ(δ) σ S n L q(z) ) β + q (Z) g Lq(Z),

9 Pisier s inequality revisited 9 It remains to note that for each δ { 1, 1} n we have (34) 1 n! σ S n 1 n! ( ) δ j g σ σ(j) gσ(j) 1 σ σ S n A A δ j j A δ j 1 j g max σ(a)σ(j) ĝ(a)w A {σ S n : max σ(a) σ(j)} δ j ĝ(a)w A n! j A δ j ĝ(a)w A A Due to (33) and (34) the proof of Theorem 21 is complete 4 Proof of Theorem 22 The following lemma introduces an auxiliary function which is a variant of a similar function that was used by Pisier in [28] Lemma 41 Let Z be a Banach space Fix n N, q [1, ] and t (, 1) For g L q (Z) define G t L q (L q (Z)) by (41) G t (δ) def 1 1 t Then (42) Rad(G t )(δ) and ĝ(a)w A A i A t A 1 j A (43) G t Lq(L q(z)) 1 tn 1 t g L q(z) (t + (1 t)δ i ) tn 1 t g δ j ĝ(a)w A, Proof Identity (42) follows from (41) since for every A {1,, n}, ( ) Rad (t + (1 t)δ i ) t A 1 (1 t) δ j i A j A

10 1 T Hytönen and A Naor To prove (43) observe that for every ε, δ { 1, 1} n, (44) (45) (1 t)g t (δ)(ε) ĝ(a)w A (ε) B {1,,n} B {1,,n} ĝ(a)w A (ε) n ( t + (1 t)δ 1 A(i) i i1 t B (1 t) n B B {1,,n} t B (1 t) n B g B (ε, δ), ) t n g(ε) t B (1 t) n B W A B (δ) t n g(ε) ĝ(a)w A B (ε)w A B (εδ) where in (44) we use (41) and in (45) for every B {1,, n} we set g B (ε, δ) def g ε j e j + ε j δ j e j j B j {1,,n} B Since g B is equidistributed with g, it follows from (45) that G t Lq(Lq(Z)) 1 t B (1 t) n B 1 tn g Lq(Z) 1 t 1 t B {1,,n} Proof of Theorem 22 Observe that for every δ { 1, 1} n we have δ j 1 1 j g δ j ĝ(a)w A A (46) It follows that if we set (47) Φ def then Φ Rad L q(z) and A A (42) Rad 1 q Φ(δ) + δ j 1 j g 1 (46) G t dt It remains to note that 1 ( 1 ( 1 j A ) t A 1 dt δ j ĝ(a)w A ) G t (δ)dt j A ( 1 ) G t dt Rad G t dt L q(z) dµ(δ) Lq(L q(z)) 1 t n dt n 1 1 t k (43) 1/q ( 1 1 tk dt n 1 k1 k ) 1 t n 1 t dt g Lq(Z)

11 Pisier s inequality revisited 11 5 Proof of Theorem 23 For t (, 1) define a linear operator V t : L q (Z) L q (L q (Z)) by (51) V t (g)(δ) def G t (δ) Ĝt( ) (41) 1 1 t ĝ(a)w A ( i A (t + (1 t)δ i ) t A ) Lemma 51 Let H be a Hilbert space Then for every t (, 1), (52) V t L2 (H) L 2 (L 2 (H)) 1 1 t t Proof Observe that for every A {1,, n} we have (53) ( ) 2 (t + (1 t)δ i ) t A dµ(δ) i A B A t 2 B (1 t) 2( A B ) ( t 2 + (1 t) 2) A t 2 A It follows from (51), (53), and the orthogonality of {W A }, that (t2 + (1 t) (54) V t L2 (H) L 2 (L 2 (H)) max 2 ) a t 2a a {1,,n} 1 t Now, for every a {1,, n} and t (, 1) we have (55) ( t 2 + (1 t) 2) a 1 a ( t 2a (1 t) 2 t 2 + (1 t) 2) a 1 k t 2k k a 1 (1 t) 2 t 2k (1 t) 2 1 t2a 1 t 1 t t, k where in the first inequality of (55) we used the estimate t 2 + (1 t) 2 1, which holds for every t [, 1] The desired estimate (52) now follows from a substitution of (55) into (54) Lemma 52 Let H be a Hilbert space and let W be a Banach space Fix θ (, 1) and q (1, ) Set Z [H, W ] θ Then for every t (, 1) we have (56) V t 2 Lq(Z) Lq(L q(z)) (1 t) 1 (1 θ) min{1/q,1 1/q}

12 12 T Hytönen and A Naor Proof For every r [1, ] we have V t (g) Lr(L r(w )) Consequently, (51) G t Ĝt( ) Lr(L r(w )) 2 G t Lr(L r(w )) (57) r [1, ], V t Lr(W ) L r(l r(w )) 2 1 t (43) 2 g L r(w ) 1 t If q [2, ) then we interpolate (see [3]) between (52) and (57) with W H and r If q (1, 2] then we interpolate between (52) and (57) with W H and r 1 The norm bound thus obtained implies the estimate (58) q (1, ), V t Lq(H) L q(l q(h)) 2 (1 t) max{1/q,1 1/q} Finally, interpolation between (58) and (57) with r q gives the desired norm bound (56) Proof of Theorem 23 By (51) we have Rad(V t (g)) Rad(G t ) Therefore, analogously to (47), if we set Ψ def 1 ( 1 ) V t (g)dt Rad G t dt 1 ( 1 ) V t (g)dt Rad V t (g)dt, then Ψ Rad L q(z) and by (46) for every δ { 1, 1} n we have (59) Ψ(δ) + Hence, δ j 1 j g Ψ(δ) + δ j 1 j g (59) (56) 1 q L q(z) 1 V t (g)(δ)dt dµ(δ) 1/q 2 g Lq(Z) dt 1 (1 θ) min{1/q,1 1/q} (1 t) 2 g Lq(Z) (1 θ) min{1/q, 1 1/q} This is precisely the assertion of Theorem 23

13 Pisier s inequality revisited 13 6 Enflo type in uniformly smooth Banach spaces A Banach space X has Rademacher type p [1, 2] (see eg [21]) if there exists T R (, ) such that for all n N and all x 1,, x n X, (61) ε j x j pdµ(ε) T p R x j p X has Enflo type p (see [9, 6, 28, 25]) if there exists T E (, ) such that for all n N and all f : { 1, 1} n X, (62) f(ε) f( ε) dµ(ε) T p { 1,1} 2 n p E p j f p L p(x) By considering the function f(ε) n ε jx j one sees that (61) is a special case of (62) It is a long-standing open problem [9] whether, conversely, (61) implies (62) A crucial feature of (62) is that it is a purely metric condition (thus one can define when a metric space has Enflo type p), while (61) is a linear condition See [22] for a purely metric condition (which is more complicated than, but inspired by, Enflo type) that is known to be equivalent to Rademacher type Observe that if (61) holds then it follows from (14) that for every f 1,, f n : { 1, 1} n X, (63) ( 1/p 1 j f T R P n p(x) f j p L p(x)) jlp(x) The special case f j j f shows that (63) implies (62) with T E T R P n p(x) For this reason it is worthwhile to investigate (63) on its own right Let Q n p(x) be the infimum over those Q (, ) such that every f 1,, f n : { 1, 1} n X satisfy (64) ( 1/p 1 j f Q f j p L p(x)) jlp(x) We also set Q p (X) def sup Q n p(x) n N

14 14 T Hytönen and A Naor By duality, Q n p(x) equals the infimum over those Q (, ) for which every g L q (X ) satisfies (65) ( 1 j g 1/q q L q(x )) Q g Lq(X ) Letting S X {x X : x 1} denote the unit sphere of X, recall that the modulus of uniform convexity of X is defined for ε [, 2] as δ X (ε) inf { x + y 1 2 } : x, y S X, x y ε The modulus of uniform smoothness of X is defined for τ (, ) as { } ρ X (τ) def x + τy + x τy sup 1 : x, y S X 2 These moduli relate to each other via the following classical duality formula of Lindenstrauss [18] { τε } (66) δ X (ε) sup 2 ρ X(τ) : τ [, 1] Theorem 61 For every K, p (1, ) there exists C(K, p) (, ) such that if X is a Banach space that satisfies ρ X (τ) Kτ p for all τ (, ), then Q p (X) C(K, p) Proof We shall use here the notation introduced in the proof of Theorem 21 (Section 3) It follows from (66) that δ X (ε) K,p ε q for every ε [, 2] (here, and it what follows, the notation K,p suppresses constant factors that may depend only on K and p) Hence, for g L q (X ) and σ S n, since {g σ k }n k, as defined in (31), is an X -valued martingale, it follows from Pisier s martingale inequality [26] that (67) ( 1/q gk σ gk 1 σ q L q(x )) K,p g Lq(X ) k1 By reindexing (67) with k σ(j), averaging over σ S n, and using the convexity of the norm, we obtain the estimate (68) 1 ( ) q g σ n! σ(j) gσ(j) 1 σ σ S n L q(x ) 1/q K,p g Lq(X )

15 Pisier s inequality revisited 15 Arguing as in (34), for every j {1,, n} we have the identity (69) 1 ( ) g σ n! σ(j) g σ 1 σ(j) 1 n! σ S n j A σ S n max σ(a)σ(j) ĝ(a)w A {σ S n : max σ(a) σ(j)} ĝ(a)w A 1 j g n! Consequently, (68) combined with (69) imply that (65) holds true with Q K,p 1 This concludes the proof of Theorem 61 Remark 61 It follows from [17, Sec 6] that a Banach space X satisfying the assumption of Theorem 61 has Enflo type p Theorem 61 can be viewed as a generalization of this fact to yield the inequality (64) In [24] it was shown that any Banach space satisfying the assumption of Theorem 61 actually has K Ball s Markov type p property [2], a property which is a useful strengthening of Enflo type p Acknowledgements T H is supported in part by the European Union through the ERC Starting Grant Analytic probabilistic methods for borderline singular integrals, and by the Academy of Finland through projects and A N is supported in part by NSF grant CCF , BSF grant 2121, the Packard Foundation and the Simons Foundation Part of this work was completed while A N was visiting Université Pierre et Marie Curie, Paris, France References [1] D J Aldous Unconditional bases and martingales in L p (F ) Math Proc Cambridge Philos Soc, 85(1): , 1979 [2] K Ball Markov chains, Riesz transforms and Lipschitz maps Geom Funct Anal, 2(2): , 1992 [3] J Bergh and J Löfström Interpolation spaces An introduction Springer-Verlag, Berlin, 1976 Grundlehren der Mathematischen Wissenschaften, No 223 [4] J Bourgain Some remarks on Banach spaces in which martingale difference sequences are unconditional Ark Mat, 21(2): , 1983

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