Pisier’s inequality revisited

Tuomas Hytönen Department of Mathematics and Statistics University of Helsinki P.O.B. 68, FI-00014 Helsinki, Finland E-mail: tuomas.hytonen@helsinki.fi

Assaf Naor Courant Institute of Mathematical Sciences New York University

251 Mercer Street, New York NY, 10012, USA E-mail: naor@cims.nyu.edu

Abstract

Given a Banach space X, for n ∈ ℕ and p ∈ (1, ∞) we investigate the smallest constant 𝔓 ∈ (0, ∞) for which every n-tuple of functions f_{1}, …, f_{n} : {−1, 1}^{n} → X satisfies

∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} ∂_{j}f_{j}(ε)∥^{p}dμ(ε)

⩽ 𝔓^{p} ∫_{{−1,1}^{n}} ∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} δ_{j}Δf_{j}(ε)∥^{p}dμ(ε)dμ(δ),

where μ is the uniform probability measure on the discrete hypercube {−1, 1}^{n} and {∂_{j}}_{j=1}^{n} and Δ = ∑_{j=1}^{n} ∂_{j} are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by 𝔓_{p}^{n}(X), we show that

𝔓_{p}^{n}(X) ⩽ ∑_{k=1}^{n} \frac{1}{k}

for every Banach space (X, ∥ · ∥). This extends the classical Pisier inequality, which corresponds to the special case f_{j} = Δ^{−1}∂_{j}f for some f : {−1, 1}^{n} → X. We show that sup_{n∈ℕ} 𝔓_{p}^{n}(X) < ∞ if either the dual X^{∗} is a UMD^{+} Banach space, or for some θ ∈ (0, 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_{p}^{n}(X) < ∞ if X is a Banach lattice of nontrivial type.

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

(1.1)  ∂_{j}f(ε) =^{def} \frac{f(ε) − f(ε_{1}, …, ε_{j−1}, −ε_{j}, ε_{j+1}, …, ε_{n})}{2}.

The hypercube Laplacian of f, denoted Δf : {−1, 1}^{n} → X, is

(1.2)  Δf(ε) =^{def} ∑_{j=1}^{n} ∂_{j}f(ε).

It is immediate to check that Δ is invertible on the space of all mean zero functions f : {−1, 1}^{n} → X. Below Δ^{−1} is understood to be defined for every f : \{−1, 1\}^{n} → X by setting Δ^{−1}f = Δ^{−1}\overline{f}. Here \overline{f} = f −∫_{\{−1,1\}^{n}} 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 ≍, i.e., A ≍ B is the same as (A ≲ B) ∧ (A ≳ B).

Theorem 1.1 (Pisier’s inequality). For every Banach space (X, ∥·∥), every n ∈ ℕ, every p ∈ [1, ∞] and every f : {−1, 1}^{n} → X, we have

(1.3) (∫_{{−1,1}^{n}} ∥f(ε) − ∫_{{−1,1}^{n}} f(δ)dμ(δ)∥^{p}dμ(ε))^{1/p}

≲ log n (∫_{{−1,1}^{n}} ∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} δ_{j}∂_{j}f(ε)∥^{p}dμ(ε)dμ(δ))^{1/p} .

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 (1.3) 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 [30] that the log n term in (1.3) is asymptotically optimal for general Banach spaces X, Wagner proved [31] that the log n term in (1.3) 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 (1.3) 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 (1.3) with 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 1.2 (Pisier constant of X). The n-dimensional Pisier constant of X (with exponent p), denoted 𝔓_{p}^{n}(X), is the infimum over those 𝔓 ∈ (0, ∞) such that every f_{1}, …, f_{n} : {−1, 1}^{n} → X satisfy

(1.4) (∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} ∂_{j}f_{j}(ε)∥^{p}dμ(ε))^{1/p}

⩽ 𝔓 (∫_{{−1,1}^{n}} ∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} δ_{j}Δf_{j}(ε)∥^{p}dμ(ε)dμ(δ))^{1/p} .

We also set

𝔓_{p}(X) =^{def} sup_{n∈ℕ} 𝔓_{p}^{n}(X).

Inequality (1.4) reduces to Pisier’s inequality if we choose f_{j} = Δ^{−1}∂_{j}f for some f : {−1, 1}^{n} → X. The generalized inequality (1.4) has the advantage of being well-behaved under duality, as explained in Section 2. The following theorem yields a logarithmic bound on 𝔓_{n}^{p}(X), thus extending Pisier’s inequality.

Theorem 1.3. For every Banach space X, every p ∈ [1, ∞] and every n ∈ ℕ,

𝔓_{p}^{n}(X) ⩽ ∑_{k=1}^{n} \frac{1}{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 (1.3) can be taken to be at most log n + O(log log n), while Theorem 1.3 shows that this term can be taken to be log n + O(1).

In [25] it was shown that the logarithmic term in (1.3) 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 ∈ (0, ∞) such that if {M_{j}}_{j=0}^{n} is a p-integrable X-valued martingale defined on some probability space (Ω, ℙ), then for every ε_{1}, …, ε_{n} ∈ {−1, 1} we have

(1.5)  ∫_{Ω} ∥M_{0} + ∑_{j=1}^{n} ε_{j}(M_{j} − M_{j−1})∥^{p}dℙ ⩽ β^{p} ∫_{Ω} ∥M_{n}∥^{p}dℙ.

The infimum over those β ∈ (0, ∞) for which (1.5) holds true is denoted β_{p}(X). It can be shown (see [7]) that β_{p}(X) ≲ \frac{p^{2}}{p−1}β_{2}(X), so in order to define the UMD property it suffices to require the validity of (1.5) for p = 2. UMD Banach spaces are known to be superreflexive [20, 1], and one also has β_{q}(X^{∗}) = β_{p}(X), where q = p/(p − 1) (see e.g. [7]).

In [10] Garling investigated the natural weakening of (1.5) 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 ∈ (0, ∞) such that if {M_{j}}_{j=0}^{n} is a p-integrable X-valued martingale defined on some probability space (Ω, ℙ) then

(1.6) ∫_{{−1,1}^{n}} ∫_{Ω} ∥M_{0} + ∑_{j=1}^{n} ε_{j}(M_{j} − M_{j−1})∥^{p}dℙdμ(ε) ⩽ β^{p} ∫_{Ω} ∥M_{n}∥^{p}dℙ.

The infimum over those β for which (1.6) holds true is denoted β_{p}^{+}(X).

Theorem 1.4. 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} ∫_{{−1,1}^{n}} δ_{j}F (ε, δ)dμ(δ).

Then

(1.7) (∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} Δ^{−1}∂_{j}F_{j}(ε)∥^{p}dμ(ε))^{1/p}
⩽ β_{q}^{+}(X^{∗}) (∫_{{−1,1}^{n}} ∫_{{−1,1}^{n}} ∥F (ε, δ)∥^{p}dμ(ε)dμ(δ))^{1/p} ,

where q = p/(p − 1).

For every f_{1}, …, f_{n} : {−1, 1}^{n} → X, an application of Theorem 1.4 to the function F (ε, δ) = ∑_{j=1}^{n} δ_{j}f_{j}(ε) yields the following estimate on the Pisier constant of a UMD^{+} Banach space.

Corollary 1.5. 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 1.4 and Corollary 1.5 are conceptually different from the result of [25], which relies on the full force of the UMD condition, i.e. it requires the validity of (1.5) 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 (1.3) replaced by β_{p}(X) = β_{q}(X^{∗}), while we obtain the same estimate with the O(log n) term in (1.3) replaced by β_{q}^{+}(X^{∗}) ⩽ β_{q}(X^{∗}). Geiss proved [11] that for every η ∈ (0, 1) there is C_{η} ∈ (0, ∞) 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 1.1. Inequality (1.7) is an extension of the generalized Pisier inequality (1.4), but for general Banach spaces it behaves very differently: unlike the logarithmic behavior of Theorem 1.3, the best constant appearing in the right hand side of (1.7) for a general Banach space X must be at least a constant multiple of \sqrt{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) = ∏_{i=1}^{n}(1 + ε_{i}δ_{i}η_{i}).

Suppose that θ ∈ (0, 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 1.6 below shows that in this case 𝔓_{p}(X) < ∞, and therefore Pisier’s inequality holds true with the log n term in (1.3) 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 ∈ (0, 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]).

Theorem 1.6. Let X, Y be Banach spaces and let H be a Hilbert space. Suppose that for some θ ∈ (0, 1) we have X = [H, Y]_{θ}. Then for every p ∈ (1, ∞),

𝔓_{p}(X) ⩽ \frac{2 max\{p, p/(p − 1)\}}{1 − θ}.

Remark 1.2. If r ∈ (2, ∞) then the O(log n) term in Pisier’s inequality (1.3), 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 1.6. At the same time, an inspection of Talagrand’s example in [30] 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 = ∑_{A⊆{1,...,n}} f̂(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 f̂(A) ∈ X is given by f̂(A) = ∫_{{−1,1}^{n}} f(x)W_{A}(x)dμ(x). Using this (standard) notation, we have

∀ i ∈ \{1, …, n\}, ∀ f ∈ L_{p}(X), ∂_{i}f = ∑_{\begin{matrix}A⊆\{1,...,n\} \\ i∈A \end{matrix}} f̂(A)W_{A},

∀ f ∈ L_{p}(X), Δf = ∑_{A⊆{1,...,n}} |A|f̂(A)W_{A},

and

∀ f ∈ L_{p}(X), Δ^{−1}f =^{def} ∑_{\begin{matrix}A⊆\{1,...,n\} \\ A≠∅ \end{matrix}} \frac{1}{|A|}f̂(A)W_{A}.

The Rademacher projection of f ∈ L_{p}(X) is defined as usual by

Rad(f) =^{def} ∑_{i=1}^{n} f̂({i})W_{{i}}.

We denote below Rad_{X} =^{def} Rad(L_{p}(X)) and Rad_{X}^{⊥} =^{def} (I − Rad)(L_{p}(X)). The dual of (Rad_{X}, ∥ · ∥_{L_{p}(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

(2.1)  ∀ F ∈ L_{p}(L_{p}(X)), S(F) =^{def} ∑_{j=1}^{n} Δ^{−1}∂_{j}F̂({j}).

Using this notation, Theorem 1.4 is nothing more than the following operator norm bound.

∥S∥_{L_{p}(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) = ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g.

Therefore Theorem 1.4 has the following equivalent dual formulation.

Theorem 2.1 (Dual formulation of Theorem 1.4). Let Z be a UMD^{+} Banach space. Then for every q ∈ (1, ∞) and every g ∈ L_{q}(Z) we have

(∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g∥_{L_{q}(Z)}^{q}dμ(δ))^{1/q} ⩽ β_{q}^{+}(Z)∥g∥_{L_{q}(Z)}.

Theorem 2.1, and consequently also Theorem 1.4, will be proven in Section 3.

Let T be the restriction of S to Rad_{L_{p}(X)}. Thus

𝔓_{p}^{n}(X) = ∥T ∥_{Rad_{Lp(X)}→L_{p}(X)} = ∥T^{∗}∥_{L_{q}(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) = ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g + Rad_{L_{q}(X^{∗})}^{⊥}.

Therefore Theorem 1.3 has the following equivalent dual formulation.

Theorem 2.2 (Dual formulation of Theorem 1.3). Let Z be a Banach space and q ∈ [1, ∞]. Then for every g ∈ L_{q}(Z) we have

inf_{Φ∈Rad_{Lq(Z)}^{⊥}} (∫_{{−1,1}^{n}} ∥Φ(δ) + ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g∥_{L_{q}(Z)}^{q}dμ(δ))^{1/q}

⩽ (∑_{k=1}^{n} \frac{1}{k}) ∥g∥_{L_{q}(Z)}.

Theorem 2.2, and consequently also Theorem 1.3, 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 1.6.

Theorem 2.3 (Dual formulation of Theorem 1.6). Let H be a Hilbert space, W a Banach space, and θ ∈ (0, 1). Set Z = [H, W]_{θ}. Then for every q ∈ (1, ∞) and g ∈ L_{q}(Z),

inf_{Ψ∈Rad_{Lq(Z)}^{⊥}} (∫_{{−1,1}^{n}} ∥Ψ(δ) + ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g∥_{L_{q}(Z)}^{q}dμ(δ))^{1/q}

⩽ \frac{2 max\{q, q/(q − 1)\}}{1 − θ}∥g∥_{L_{q}(Z)}.

Theorem 2.3, and consequently also Theorem 1.6, will be proven in Section 5.

3 Proof of Theorem 2.1

Fix q ∈ (1, ∞) and g ∈ L_{q}(Z). Let S_{n} denote the symmetric group on {1, …, n}. For σ ∈ S_{n} and k ∈ {0, …, n} define g_{k}^{σ} ∈ L_{q}(Z) by

(3.1) g_{k}^{σ}(ε) =^{def} ∑_{A⊆{σ^{−1}(1),...,σ^{−1}(k)}} ĝ(A)W_{A}(ε)

= \frac{1}{2^{n−k}} ∑_{δ_{σ^{−1}(k+1)},...,δ_{σ^{−1}(n)}∈\{−1,1\}} g (∑_{i=1}^{k} ε_{σ^{−1}(i)}e_{σ^{−1}(i)} + ∑_{i=k+1}^{n} 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 {g_{k}^{σ}}_{k=0}^{n} is a Z-valued martingale with g_{n}^{σ} = g and g_{0}^{σ} = ĝ(∅), implying that

(3.2) (∫_{{−1,1}^{n}} ∥ ∑_{k=1}^{n} δ_{k} (g_{k}^{σ} − g_{k−1}^{σ}) ∥_{L_{q}(Z)}^{q}dμ(δ))^{1/q} ⩽ β_{q}^{+}(Z) ∥g∥_{L_{q}(Z)} .

In (3.2) we may replace {δ_{k}}_{k=1}^{n} by {δ_{σ^{−1}(k)}}_{k=1}^{n}, 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

(3.3) (∫_{\{−1,1\}^{n}} ∥\frac{1}{n!} ∑_{σ∈S_{n}} ∑_{j=1}^{n} δ_{j} (g_{σ(j)}^{σ} − g_{σ(j)−1}^{σ}) ∥_{L_{q}(Z)}^{q}dμ(δ))^{1/q}

⩽ β_{q}^{+}(Z) ∥g∥_{L_{q}(Z)} .

It remains to note that for each δ ∈ {−1, 1}^{n} we have

\frac{1}{n!} ∑_{σ∈S_{n}} ∑_{j=1}^{n} δ_{j} (g_{σ(j)}^{σ} − g_{σ(j)−1}^{σ})

= \frac{1}{n!} ∑_{σ∈S_{n}} ∑_{j=1}^{n} δ_{j} ∑_{\begin{matrix}∅⊊A⊆\{1,...,n\} \\ max σ(A)=σ(j) \end{matrix}} ĝ(A)W_{A}

= ∑_{\begin{matrix}A⊆\{1,...,n\} \\ A≠∅ \end{matrix}} ∑_{j∈A} δ_{j} \frac{|\{σ ∈ S_{n} : max σ(A) = σ(j)\}|}{n!}ĝ(A)W_{A}

= ∑_{\begin{matrix}A⊆\{1,...,n\} \\ A≠∅ \end{matrix}} \frac{∑_{j∈A} δ_{j}}{|A|} ĝ(A)W_{A}

= ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g. (3.4)

Due to (3.3) and (3.4) the proof of Theorem 2.1 is complete.

4 Proof of Theorem 2.2

The following lemma introduces an auxiliary function which is a variant of a similar function that was used by Pisier in [28].

Lemma 4.1. Let Z be a Banach space. Fix n ∈ ℕ, q ∈ [1, ∞] and t ∈ (0, 1). For g ∈ L_{q}(Z) define G_{t} ∈ L_{q}(L_{q}(Z)) by

(4.1)  G_{t}(δ) =^{def} \frac{1}{1 − t} ∑_{A⊆\{1,...,n\}} ĝ(A)W_{A} ∏_{i∈A} (t + (1 − t)δ_{i}) − \frac{t^{n}}{1 − t}g.

Then

(4.2)  Rad(G_{t})(δ) = ∑_{\begin{matrix}A⊆\{1,...,n\} \\ A≠∅ \end{matrix}} t^{|A|−1} ∑_{j∈A} δ_{j}ĝ(A)W_{A},

and

(4.3)  ∥G_{t}∥_{L_{q}(L_{q}(Z))} ⩽ \frac{1 − t^{n}}{1 − t} ∥g∥_{L_{q}(Z)}.

Proof. Identity (4.2) follows from (4.1) since for every A ⊆ {1, …, n},

Rad (∏_{i∈A} (t + (1 − t)δ_{i})) = t^{|A|−1}(1 − t) ∑_{j∈A} δ_{j}.

10 T. Hytönen and A. Naor To prove (4.3) observe that for every ε, δ ∈ {−1, 1}^{n},

(1 − t)G_{t}(δ)(ε)

= ∑_{A⊆{1,...,n}} ĝ(A)W_{A}(ε) ∏_{i=1}^{n} (t + (1 − t)δ_{i}^{1_{A}(i)}) − t^{n}g(ε) (4.4)

= ∑_{A⊆{1,...,n}} ĝ(A)W_{A}(ε) ∑_{B⊆{1,...,n}} t^{|B|}(1 − t)^{n−|B|}W_{A∖B}(δ) − t^{n}g(ε)

= ∑_{B⊊{1,...,n}} t^{|B|}(1 − t)^{n−|B|} ∑_{A⊆{1,...,n}} ĝ(A)W_{A∩B}(ε)W_{A∖B}(εδ)

= ∑_{B⊊{1,...,n}} t^{|B|}(1 − t)^{n−|B|}g_{B}(ε, δ), (4.5)

where in (4.4) we use (4.1) and in (4.5) for every B ⊆ {1, …, n} we set

g_{B}(ε, δ) =^{def} g (∑_{j∈B} ε_{j}e_{j} + ∑_{j∈{1,...,n}∖B} ε_{j}δ_{j}e_{j}) .

Since g_{B} is equidistributed with g, it follows from (4.5) that

\frac{∥G_{t}∥_{L_{q}(L_{q}(Z))}}{∥g∥_{L_{q}(Z)}} ⩽ \frac{1}{1 − t} ∑_{B⊊\{1,...,n\}} t^{|B|}(1 − t)^{n−|B|} = \frac{1 − t^{n}}{1 − t} .

Proof of Theorem 2.2. Observe that for every δ ∈ {−1, 1}^{n} we have

∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g = ∑_{\begin{matrix}A⊆\{1,...,n\} \\ A≠∅ \end{matrix}} \frac{1}{|A|} ∑_{j∈A} δ_{j}ĝ(A)W_{A}
= ∑_{\begin{matrix}A⊆\{1,...,n\} \\ A≠∅ \end{matrix}} (∫_{0}^{1} t^{|A|−1}dt) ∑_{j∈A} δ_{j}ĝ(A)W_{A}
(4.6)  =^{(4.2)} Rad (∫_{0}^{1} G_{t}(δ)dt) .

It follows that if we set

(4.7)  Φ =^{def} ∫_{0}^{1} G_{t}dt − Rad (∫_{0}^{1} G_{t}dt) .

then Φ ∈ Rad_{L_{q}(Z)}^{⊥} and

(∫_{{−1,1}^{n}} ∥Φ(δ) + ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g∥_{L_{q}(Z)}^{q}dμ(δ))^{1/q}

=^{(4.6)} ∥ ∫_{0}^{1} G_{t}dt∥_{L_{q}(L_{q}(Z))} ⩽^{(4.3)} (∫_{0}^{1} \frac{1 − t^{n}}{1 − t} dt) ∥g∥_{L_{q}(Z)}.

It remains to note that ∫_{0}^{1} \frac{1−t^{n}}{1−t} dt = ∑_{k=0}^{n−1} ∫_{0}^{1} t^{k}dt = ∑_{k=1}^{n} \frac{1}{k}.

5 Proof of Theorem 2.3

For t ∈ (0, 1) define a linear operator V_{t} : L_{q}(Z) → L_{q}(L_{q}(Z)) by

(5.1) V_{t}(g)(δ) =^{def} G_{t}(δ) − Ĝ_{t}(∅)

=^{(4.1)} \frac{1}{1 − t} ∑_{A⊆\{1,...,n\}} ĝ(A)W_{A} (∏_{i∈A}(t + (1 − t)δ_{i}) − t^{|A|}) .

Lemma 5.1. Let H be a Hilbert space. Then for every t ∈ (0, 1),

(5.2)  ∥V_{t}∥_{L_{2}(H)→L_{2}(L_{2}(H))}⩽\frac{1}{\sqrt{1 − t^{2}}} ⩽ \frac{1}{\sqrt{1 − t}}.

Proof. Observe that for every A ⊆ {1, …, n} we have

(5.3) ∫_{{−1,1}^{n}} (∏_{i∈A}(t + (1 − t)δ_{i}) − t^{|A|})^{2} dμ(δ)

= ∑_{B⊊A} t^{2|B|}(1 − t)^{2(|A|−|B|)} = (t^{2} + (1 − t)^{2})^{|A|} − t^{2|A|}.

It follows from (5.1), (5.3), and the orthogonality of {W_{A}}_{A⊆{1,...,n}}, that

(5.4)  ∥V_{t}∥_{L_{2}(H)→L_{2}(L_{2}(H))} = max_{a∈\{1,...,n\}} \frac{\sqrt{(t^{2} + (1 − t)^{2})^{a} − t^{2a}}}{1 − t} .

Now, for every a ∈ {1, …, n} and t ∈ (0, 1) we have

(5.5) (t^{2} + (1 − t)^{2})^{a} − t^{2a} = (1 − t)^{2} ∑_{k=0}^{a−1} (t^{2} + (1 − t)^{2})^{a−1−k} t^{2k}

⩽ (1 − t)^{2} ∑_{k=0}^{a−1} t^{2k} = (1 − t)^{2} \frac{1 − t^{2a}}{1 − t^{2}} ⩽ \frac{1 − t}{1 + t},

where in the first inequality of (5.5) we used the estimate t^{2} + (1 − t)^{2} ⩽ 1, which holds for every t ∈ [0, 1]. The desired estimate (5.2) now follows from a substitution of (5.5) into (5.4).

Lemma 5.2. Let H be a Hilbert space and let W be a Banach space. Fix θ ∈ (0, 1) and q ∈ (1, ∞). Set Z = [H, W]_{θ}. Then for every t ∈ (0, 1) we have

(5.6)  ∥V_{t}∥_{L_{q}(Z)→L_{q}(L_{q}(Z))} ⩽ \frac{2}{(1 − t)^{1−(1−θ) min\{1/q,1−1/q\}}} .

Proof. For every r ∈ [1, ∞] we have

∥V_{t}(g)∥_{L_{r}(L_{r}(W))} =^{(5.1)} ∥G_{t} − Ĝ_{t}(∅)∥_{L_{r}(L_{r}(W))}

⩽ 2∥G_{t}∥_{L_{r}(L_{r}(W))} ⩽^{(4.3)} \frac{2∥g∥_{L_{r}(W)}}{1 − t} .

Consequently,

(5.7)  ∀ r ∈ [1, ∞], ∥V_{t}∥_{L_{r}(W)→L_{r}(L_{r}(W))} ⩽ \frac{2}{1 − t}.

If q ∈ [2, ∞) then we interpolate (see [3]) between (5.2) and (5.7) with W = H and r = ∞. If q ∈ (1, 2] then we interpolate between (5.2) and (5.7) with W = H and r = 1. The norm bound thus obtained implies the estimate

(5.8)  ∀ q ∈ (1, ∞), ∥V_{t}∥_{L_{q}(H)→L_{q}(L_{q}(H))} ⩽ \frac{2}{(1 − t)^{max\{1/q,1−1/q\}}} .

Finally, interpolation between (5.8) and (5.7) with r = q gives the desired norm bound (5.6).

Proof of Theorem 2.3. By (5.1) we have Rad(V_{t}(g)) = Rad(G_{t}). Therefore, analogously to (4.7), if we set

Ψ =^{def} ∫_{0}^{1} V_{t}(g)dt − Rad (∫_{0}^{1} G_{t}dt) = ∫_{0}^{1} V_{t}(g)dt − Rad (∫_{0}^{1} V_{t}(g)dt) ,

then Ψ ∈ Rad_{L_{q}(Z)}^{⊥} and by (4.6) for every δ ∈ {−1, 1}^{n} we have

(5.9)  Ψ(δ) + ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g = ∫_{0}^{1} V_{t}(g)(δ)dt.

Hence,

(∫_{{−1,1}^{n}} ∥Ψ(δ) + ∑_{j=1}^{n} δ_{j}Δ^{−1}∂_{j}g∥_{L_{q}(Z)}^{q}dμ(δ))^{1/q}
⩽^{(5.9)∧(5.6)} ∫_{0}^{1} \frac{2∥g∥_{L_{q}(Z)}}{(1 − t)^{1−(1−θ) min\{1/q,1−1/q\}}} dt
=  \frac{2∥g∥_{L_{q}(Z)}}{(1 − θ) min\{1/q, 1 − 1/q\}}.

This is precisely the assertion of Theorem 2.3.

6 Enflo type in uniformly smooth Banach spaces

A Banach space X has Rademacher type p ∈ [1, 2] (see e.g. [21]) if there exists T_{R} ∈ (0, ∞) such that for all n ∈ ℕ and all x_{1}, …, x_{n} ∈ X,

(6.1)  ∫_{{−1,1}^{n}} ∥ ∑_{j=1}^{n} ε_{j}x_{j}∥^{p}dμ(ε) ⩽ T_{R}^{p} ∑_{j=1}^{n} ∥x_{j}∥^{p}.

X has Enflo type p (see [9, 6, 28, 25]) if there exists T_{E} ∈ (0, ∞) such that for all n ∈ ℕ and all f : {−1, 1}^{n} → X,

(6.2)  ∫_{\{−1,1\}^{n}} \frac{∥f(ε) − f(−ε)∥^{p}}{2^{p}} dμ(ε) ⩽ T_{E}^{p} ∑_{j=1}^{n} ∥∂_{j}f∥_{L_{p}(X)}^{p}.

By considering the function f(ε) = ∑_{j=1}^{n} ε_{j}x_{j} one sees that (6.1) is a special case of (6.2). It is a long-standing open problem [9] whether, conversely, (6.1) implies (6.2). A crucial feature of (6.2) is that it is a purely metric condition (thus one can define when a metric space has Enflo type p), while (6.1) 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 (6.1) holds then it follows from (1.4) that for every f_{1}, …, f_{n} : {−1, 1}^{n} → X,

(6.3)  ∥ ∑_{j=1}^{n} Δ^{−1}∂_{j}f_{j}∥_{L_{p}(X)} ⩽ T_{R}𝔓_{p}^{n}(X) (∑_{j=1}^{n} ∥f_{j}∥_{L_{p}(X)}^{p})^{1/p} .

The special case f_{j} = ∂_{j}f shows that (6.3) implies (6.2) with

T_{E} ⩽ T_{R}𝔓_{p}^{n}(X).

For this reason it is worthwhile to investigate (6.3) on its own right.

Let 𝔔_{p}^{n}(X) be the infimum over those 𝔔 ∈ (0, ∞) such that every f_{1}, …, f_{n} : {−1, 1}^{n} → X satisfy

(6.4)  ∥ ∑_{j=1}^{n} Δ^{−1}∂_{j}f_{j}∥_{L_{p}(X)} ⩽ 𝔔 (∑_{j=1}^{n} ∥f_{j}∥_{L_{p}(X)}^{p})^{1/p} .

We also set

𝔔_{p}(X) =^{def} sup_{n∈ℕ} 𝔔_{p}^{n}(X).

By duality, 𝔔_{p}^{n}(X) equals the infimum over those 𝔔 ∈ (0, ∞) for which every g ∈ L_{q}(X^{∗}) satisfies

(6.5)  (∑_{j=1}^{n} ∥Δ^{−1}∂_{j}g∥_{L_{q}(X^{∗})}^{q})^{1/q} ⩽ 𝔔∥g∥_{L_{q}(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 ∈ [0, 2] as

δ_{X}(ε) = inf \{1 − \frac{∥x + y∥}{2} : x, y ∈ S_{X}, ∥x − y∥ = ε\} .

The modulus of uniform smoothness of X is defined for τ ∈ (0, ∞) as

ρ_{X}(τ) =^{def} sup \{\frac{∥x + τy∥+∥x − τy∥}{2} − 1 : x, y ∈ S_{X}\} .

These moduli relate to each other via the following classical duality formula of Lindenstrauss [18].

(6.6)  δ_{X^{∗}}(ε) = sup \{\frac{τε}{2} − ρ_{X}(τ) : τ ∈ [0, 1]\} .

Theorem 6.1. For every K, p ∈ (1, ∞) there exists C(K, p) ∈ (0, ∞) such that if X is a Banach space that satisfies ρ_{X}(τ) ⩽ Kτ^{p} for all τ ∈ (0, ∞), then 𝔔_{p}(X) ⩽ C(K, p).

Proof. We shall use here the notation introduced in the proof of Theorem 2.1 (Section 3). It follows from (6.6) that δ_{X^{∗}}(ε)≳_{K,p}ε^{q} for every ε ∈ [0, 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_{k}^{σ}}_{k=0}^{n}, as defined in (3.1), is an X^{∗}-valued martingale, it follows from Pisier’s martingale inequality [26] that

(6.7)  (∑_{k=1}^{n} ∥g_{k}^{σ} − g_{k−1}^{σ}∥_{L_{q}(X^{∗})}^{q})^{1/q} ≲_{K,p} ∥g∥_{L_{q}(X^{∗})}.

By reindexing (6.7) with k = σ(j), averaging over σ ∈ S_{n}, and using the convexity of the norm, we obtain the estimate

(6.8)  (∑_{j=1}^{n} ∥\frac{1}{n!} ∑_{σ∈S_{n}} (g_{σ(j)}^{σ} − g_{σ(j)−1}^{σ}) ∥_{L_{q}(X^{∗})}^{q})^{1/q} ≲_{K,p} ∥g∥_{L_{q}(X^{∗})}.

Arguing as in (3.4), for every j ∈ {1, …, n} we have the identity

(6.9) \frac{1}{n!} ∑_{σ∈S_{n}} (g_{σ(j)}^{σ} − g_{σ(j)−1}^{σ}) = \frac{1}{n!} ∑_{σ∈S_{n}} ∑_{\begin{matrix}∅⊊A⊆\{1,...,n\} \\ max σ(A)=σ(j) \end{matrix}} ĝ(A)W_{A}

= ∑_{\begin{matrix}A⊆\{1,...,n\} \\ j∈A \end{matrix}} \frac{|\{σ ∈ S_{n} : max σ(A) = σ(j)\}|}{n!}ĝ(A)W_{A} = Δ^{−1}∂_{j}g.

Consequently, (6.8) combined with (6.9) imply that (6.5) holds true with 𝔔 ≲_{K,p} 1. This concludes the proof of Theorem 6.1.

Remark 6.1. It follows from [17, Sec. 6] that a Banach space X satisfying the assumption of Theorem 6.1 has Enflo type p. Theorem 6.1 can be viewed as a generalization of this fact to yield the inequality (6.4). In [24] it was shown that any Banach space satisfying the assumption of Theorem 6.1 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 130166 and 133264. A. N. is supported in part by NSF grant CCF-0832795, BSF grant 2010021, 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.

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