ON SINGULAR INTEGRAL AND MARTINGALE TRANSFORMS

STEFAN GEISS, STEPHEN MONTGOMERY-SMITH, AND EERO SAKSMAN

Abstract. Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on L_{X}^{p}(R^{2}) with p ∈ (1, ∞). Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals p^{∗} − 1 with p^{∗} := max{p, (p/(p − 1))}, where the novelty is the lower bound.

1. Introduction

A Banach space X is said to be a UMD-space ^{1} provided that for all (equivalently, for some) p ∈ (1, ∞) there is a constant c_{p} > 0 such that

sup_{α_{k}∈{±1}} ∥∑_{k=1}^{n} α_{k}D_{k}∥_{L_{X}^{p}} ≤ c_{p} ∥∑_{k=1}^{n} D_{k}∥_{L_{X}^{p}}

for all n ≥ 1 and all X-valued martingale difference sequences (D_{k})_{k=1}^{n}. As UMD-constant one usually takes UMD_{p}(X) := inf c_{p}. We refer to [11] and the references therein for an overview about the UMD-property. It is known that in the above definition the arbitrary martingale differences can be replaced by Walsh-Paley martingale differences and one gets the same constant (see e.g. [10, p. 12] and [20]; the definition of Walsh-Paley martingales is recalled in Section 2 below). The UMD-property was first investigated by Burkholder [7], who gave a geometric characterization for Banach spaces with the UMD-property. Together with McConnell, Burkholder established in [8] that the Hilbert transform

H f(x) := \frac{1}{π} ∫_{R} \frac{f(y)}{x − y} dy (1)

is bounded on L_{X}^{p}(R), p ∈ (1, ∞), provided that X has the UMD-property. A converse result was proved soon after by Bourgain [5], who showed that the boundedness of the Hilbert transform on L_{X}^{p} for some p ∈ (1, ∞) implies the UMD-property for X. It is also known from [6] that the UMD-property implies the boundedness of all invariant singular integrals or (more generally) standard multiplier operators under some regularity assumptions. The importance of the UMD-property, especially in connection with PDE:s, is further evidenced by recent new results on operator valued singular integrals (R-boundedness) [26] and other developments.

2000 Mathematics Subject Classification. 60G46, 42B15 (Primary), 42B20, 46B09, 46B20 (Secondary).

Key words and phrases. UMD property, singular integrals, martingale transforms.

The first and the last author are supported by the Project #110599 of the Academy of Finland.

1UMD stands for unconditional martingale differences.

It is natural to ask for the quantitative equivalence of the UMD-property and the boundedness of the vector-valued Hilbert transform. The proofs in [8] and [5] yield that there is a constant C > 0 such that

(2) \frac{1}{C} (UMD_{2}(X))^{1/2} ≤ ∥ H ∥_{L_{X}^{2}(R)→L_{X}^{2}(R)} ≤ C(UMD_{2}(X))^{2}.

The curious feature above is the quadratic equivalence of the norms in contrast to the linear dependence one would expect. A better than a quadratic equivalence obtained from alternative proofs is not known to the authors.

The previous discussion raises the question whether there is a linear equivalence in (2). We do not know whether this is true or not. However, in this paper we establish that the answer is positive if H is replaced by the Beurling-Ahlfors transform BA:

Theorem 1.1. For p ∈ (1, ∞) and a real Banach space X one has that

(3) UMD_{p}(X) = ∥Re(BA)∥_{L_{X}^{p}(R^{2})→L_{X}^{p}(R^{2})} = ∥Im(BA)∥_{L_{X}^{p}(R^{2})→L_{X}^{p}(R^{2})}

with

BA f(z) := −\frac{1}{π} ∫_{C} \frac{f(w)}{(z − w)^{2}} dm_{2}(w), (4)

where m_{2} is the two-dimensional Lebesgue measure on the complex plane C, that is identified with R^{2}.

Note that Re(BA) = −Id − 2R_{1}^{2} and Im(BA) = 2R_{1}R_{2} where R_{1} and R_{2} are the first and second Riesz transform, respectively. Equality (3) carries some new information for norm estimates even in the scalar case, see Corollary 3.2 below and the remark after it.

The operator BA is sometimes called the two-dimensional Hilbert transform. It plays a fundamental role in the theory of quasi-conformal maps and in the theory of elliptic equations in the plane. Quite recently, in connection with the well-known Iwaniec conjecture, there has been many works devoted to the probabilistic approach to estimate the (scalar) L^{p}-norm of the Beurling-Ahlfors operator, see e.g. [1, 25, 2].

If we replace the equality in (3) by a linear equivalence with multiplicative constants, then the obtained property is shared by an extensive class of operators corresponding to homogeneous multipliers. In this context the crucial difference between the Hilbert transform and the Beurling-Ahlfors operator is the parity of their integral kernels: our main Theorem 3.1 shows that one may replace the real and imaginary part of BA in (3) by any smooth, homogeneous of order zero, and even Fourier multiplier operator if we allow multiplicative constants. On the other side, Theorem 4.1 in turn relates odd multipliers to the Hilbert transform.

The proof of Theorem 1.1 is based on a modification of an argument of Bourgain and the already standard representation of suitable combinations of Riesz singular integrals as certain transforms of Ito-integrals, see [16, 1]. In order to define a (wider) class of such transforms, let W = ((W_{t}^{1}, ..., W_{t}^{d}))_{t≥0} be a standard d-dimensional Brownian motion. For a Banach space X, p ∈ (1, ∞), and a real d × d matrix A we let ∥Id_{X}|(A, W)∥_{p} := inf c, where the infimum is taken over all c > 0 such that

∥ ∫_{0}^{T} U · d(AW_{t})∥_{L_{X}^{p}} ≤ c∥ ∫_{0}^{T} U · d(W_{t})∥_{L_{X}^{p}}

for all T ≥ 0 and certain d-tuples U of X-valued processes taking values in a finite-dimensional subspace of X (see Section 2). As a byproduct of our proofs we obtain in Theorem 3.1 that, if the matrix A is symmetric and not a constant multiple of the identity matrix, then there is a constant C = C(A) > 0, independent of the Banach space X, such that

\frac{1}{C} UMD_{p}(X) ≤ ∥Id_{X}|(A, W)∥_{p} ≤ C UMD_{p}(X). (5)

In the proof of Theorem 1.1 (in fact, Theorem 3.1) we do not employ harmonic extensions of functions to the upper half plane as it was done originally by Gundy and Varopoulos [16] in their stochastic proof of the L^{p}-boundedness of the scalar Riesz transforms. Instead, we follow Bañuelos and Mendez-Hernandez [2] and use the space-time Brownian motion, which corresponds to the heat extension of functions. Earlier the use of heat extensions in norms estimates for singular integrals was initiated by Nazarov, Petermichl and Volberg in [21] and [25]. The use of the space-time Brownian motion in combination with our modified version of Bourgain techniques is one reason that enables us to obtain the equalities (3). In particular, for all d ≥ 2 it holds that

(6)  UMD_{p}(X) = ∥Id + 2R_{1}^{2}∥_{L_{X}^{p}(R^{d})→L_{X}^{p}(R^{d})} = ∥Id_{X}|(A_{s}, W)∥_{p}

with

A_{s} := [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}].

As one specializes to the case X = R in Theorem 1.1 one obtains

Corollary 1.2. For p ∈ (1, ∞) one has that

∥Re(BA)∥_{L^{p}(R^{2})→L^{p}(R^{2})} = p^{∗} − 1 where p^{∗} := max (p, \frac{p}{p − 1}) .

This follows because Burkholder (see [9], [11, Theorem 14], [10, p.12]) established that UMD_{p}(R) = p^{∗} − 1. One should also observe here, that it is well-known that the L^{p}−norm of scalar multipliers that preserve real-valued functions does not depend on whether one look at real or complex L^{p}-spaces. The upper bound in Corollary 1.2 is due to Volberg and Nazarov [25, Theorem 3.1] who applied Bellman function and heat kernel techniques in the proof (a space-time Brownian motion approach to [25, Theorem 3.1] was subsequently given in [2]). The real novelty in our Corollary 1.2 is the lower bound. As far as we know, this is the first instance where the exact value of the norm of a ’second order’ Riesz transform (a non-trivial expression involving products of Riesz transforms) has been determined. Moreover, Corollary 1.2 has obvious interest in connection with the famous Iwaniec conjecture, which claims that the norm of the full complex Beurling-Ahlfors transform equals p^{∗} − 1.

The paper is organized as follows: Section 2 recalls necessary notation and contains preparatory material. Our main result, Theorem 3.1, which contains Theorem 1.1, (5), and (6) as special cases, is formulated and proved in Section 3. Section 4 treats the case of odd kernels and anti-symmetric matrix transforms of stochastic integrals.

The results in Sections 3 and 4 are formulated for tensor product operators T_{m} ⊗ S, where S : X → Y is an arbitrary operator between two Banach spaces X and Y, instead for the setting of identities Id_{X} we have used so far in the introduction. The motivation for this are certain connections to the geometry of Banach spaces that are explained in Section 5.

We would like to thank Tuomas Hytönen for the careful reading of the manuscript. In his recent paper [17] he obtained results about the linear equivalence of norms of vector valued spectral multipliers and the UMDconstant, which are in the spirit of the results in this paper (but, as pointed out in [17], the basic singular integral operators like BA or H do not fall into the setting and scope of [17]).

2. Definitions and preliminary results

We shall denote by {e_{1}, …, e_{d}} the unit vectors of R^{d}, by |x| the euclidean norm of x ∈ R^{d}, and will use B(x, δ) := {y ∈ R^{d} : |x − y| ≤ δ} for δ ≥ 0. Moreover, the set of real m × n matrices is denoted by M(m, n).

Vector-valued operators. In this paper L(X, Y) stands for the linear and bounded operators between two real Banach spaces X and Y, where L(X) := L(X, X). Given a σ-finite measure space (M, μ) and p ∈ [1, ∞), the space of Bochner integrable random variables L_{X}^{p}(M) = L_{X}^{p}(M, μ) consists of all strongly measurable functions f : M → X such that there is a separable subspace X_{0} ⊆ X with f(M) ⊆ X_{0} and ∥f∥_{L_{X}^{p}} := (∫_{M} ∥f∥^{p}dμ)^{1/p} < ∞. Given operators S ∈ L(X, Y) and T ∈ L(L^{p}(M, μ)) the tensor product T ⊗S can be defined through its action on simple functions, i.e.

(T ⊗ S) (∑_{k=1}^{n} x_{k}χ_{E_{k}}) := ∑_{k=1}^{n} Sx_{k}T (χ_{E_{k}}).

In case that the quantity

∥T ⊗ S : L_{X}^{p}(M, μ) → L_{Y}^{p} (M, μ)∥

:= sup {∥(T ⊗ S)(F)∥_{L_{Y}^{p}} : ∥F ∥_{L_{X}^{p}} ≤ 1, F simple function}

is finite, the operator T ⊗ S extends to a bounded linear operator from L_{X}^{p}(M, μ) into L_{Y}^{p} (M, μ). As particular operators T we use multipliers. The usage of operators S instead of identities Id : X → X of Banach spaces X might be seen as somehow artificial at this point. As already mentioned, the motivation for this slightly more general setting can be found in Section 5, but the reader may, if she or he so wishes, replace the operator S by an identity in what follows.

Multipliers. A bounded complex valued function m ∈ C^{∞}(R^{d}\{0}), d ≥ 1, is called (smooth) multiplier. A multiplier m is homogeneous (of order zero) if m(λξ) = m(ξ) for ξ ∈ R^{d} \ {0} and λ > 0. In this paper the term multiplier always refers to smooth and homogeneous multipliers. The multiplier m is called even provided that m(ξ) = m(−ξ) for ξ ≠ 0 and odd if m(ξ) = −m(−ξ). The operator T_{m} : L^{2}(R^{d}) → L^{2}(R^{d}) associated to m is given by

T_{m}f := F^{−1}(mF f),

where F stands for the Fourier transform

(F f)(ξ) := ∫_{R^{d}} exp(−i⟨ξ, x⟩)f(x)dx.

It is easy to check that an even and real multiplier maps real-valued functions to real-valued functions, and the same is true for odd and purely imaginary multipliers. Consequently, to formulate our results for real Banach spaces, from now on we use the standing

Assumption 2.1. All multipliers are even and real, or odd and purely imaginary.

Let A ∈ M(d, d) be invertible. By applying the simple identity

T_{m∘A}f(x) = (T_{m}(f ∘ A^{T})) ((A^{T})^{−1}x)

we deduce that composing a multiplier with a linear invertible map one does not change its norm, i.e.

(7) ∥T_{m∘A} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ = ∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥.

For a multiplier m on R^{d} as above there is a corresponding discrete multiplier m̃ that acts on functions defined on the d-dimensional torus T^{d} := (−π, π]^{d}: for a finite trigonometric polynomial f we let

(Tm̃f)(θ) := ∑_{k∈Z^{d}} f̂(k)e^{i⟨k,θ⟩}m(k),

where f̂(k) := (1/2π)^{d} ∫_{T^{d}} e^{−i⟨k,θ⟩}f(θ)dθ and m(0) := ω_{d−1}^{−1} ∫_{S^{d−1}} m(x)dx is the average over the boundary of the euclidean unit ball (remember that m is homogeneous of order zero). It follows by Assumption 2.1 that in the above definition Tm̃f is real whenever f is real. In the sequel it will be important that the norms of the corresponding multipliers are equal which will be stated in Lemma 2.2 below.

Lemma 2.2. Let m be a smooth and homogeneous multiplier on R^{d}, d ≥ 1. Then, for any p ∈ (1, ∞) and S ∈ L(X, Y), it holds that

∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ = ∥Tm̃ ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥
= ∥Tm̃ ⊗ S : L_{X,0}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥,

where L_{X,0}^{p}(T^{d}) stands for the functions in L_{X}^{p}(T^{d}) of mean zero.

Proof. The first equality in the scalar case is essentially due to K. de Leeuw [19], see also [12]. The proof in the monograph [15, pp. 221–223] can be easily verified to carry over to the case considered here to yield the estimate

∥Tm̃ ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥ ≤ ∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥.

The lemma follows as soon as one has

∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ≤ ∥Tm̃ ⊗ S : L_{X,0}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥.

Observe that when computing the norm of T_{m} ⊗S we may restrict ourselves to functions of type f = ∑_{k=1}^{n} f_{k}x_{k} where x_{k} ∈ X and the f_{k} ∈ C_{0}^{∞}(R^{d}) have integral zero because these functions f are dense in L_{X}^{p}(R^{d}). Then we can also follow the second part of the proof of the transference principle in [15] on pp. 223–225. One also verifies, that in the above proofs one may restrict to real valued scalar functions f_{k}, and as our multipliers preserve real-valued functions, the result remains true also for real Banach spaces X, Y.

Hilbert transform. The Hilbert transform was defined in (1). It corresponds to the multiplier m(ξ) = −i sgn (ξ), and maps, by definition, real valued functions to real valued functions. The corresponding discrete multiplier operator Tm̃ is the well known conjugation operator H̃ : L_{p}(T) → L_{p}(T), p ∈ (1, ∞), which can be also defined through its action on the trigonometric polynomials

(H̃ sin(k·))(θ) := − cos(kθ) and (H cos(k·))(θ) := sin(kθ)

for k = 1, 2, ... and H̃1 :≡ 0.

Beurling-Ahlfors transform and Riesz transforms. The k-th Riesztransform R_{k}, k = 1, …, d is the multiplier operator on R^{d} corresponding to the multiplier ξ_{k}/(i|ξ|). The Beurling-Ahlfors operator, defined through

(4) in the introduction, corresponds to the multiplier m_{BA}(ξ) := \frac{ξ_{1} − iξ_{2}}{ξ_{1} + iξ_{2}} .

It follows that

BA = R_{2}^{2} − R_{1}^{2} + 2iR_{1}R_{2} =: Re(BA) + iIm(BA).

For our real valued setting we consider Re(BA) and Im(BA) separately and check by (7) that a rotation of the coordinates by the angle π/4 transforms Re(BA) into Im(BA) and their norms coincide. Martingale transforms. Given independent Bernoulli random variables ε_{1}, ε_{2}, …, i.e. P(ε_{k} = ±1) = 1/2, and maps d_{k} : R^{k−1} → X, where X is a Banach space and d_{1} is constant, a sequence (ε_{k}d_{k}(ε_{1}, ..., ε_{k−1}))_{k∈I} with I = {1, ..., n} or I = {1, 2, ...} is called Walsh-Paley martingale difference sequence. Given S ∈ L(X, Y) and p ∈ (1, ∞), we define UMD_{p}(S) := inf c, where the infimum is taken over all c > 0 such that

∥∑_{k=1}^{n} α_{k}Sε_{k}d_{k}∥_{L_{Y}^{p}} ≤ c ∥∑_{k=1}^{n} ε_{k}d_{k}∥_{L_{X}^{p}} (8)

for all X-valued Walsh-Paley martingale difference sequences (ε_{k}d_{k})_{k=1}^{n}, all α_{1}, ..., α_{n} ∈ R with |α_{k}| ≤ 1, and n ≥ 1. It is well-known that by an easy extreme point argument the condition |α_{k}| ≤ 1 can be replaced by α_{k} ∈ {−1, 1} and one gets the same constant UMD_{p}(S). The reader is also referred to [11] and the references therein for a more general overview about UMD-spaces.

Transforms for stochastic integrals. We recall the definition given in the introduction. Let W = ((W_{t}^{1}, ..., W_{t}^{d}))_{t≥0} be a d-dimensional standard Brownian motion with continuous paths for all ω ∈ Ω and W_{0} ≡ 0, defined on a probability space (Ω, F, ℙ), where F is the completion of σ(W_{t} : t ≥ 0) and (F_{t})_{t≥0} the augmentation of the natural filtration of W. Let S ∈ L(X, Y), p ∈ (1, ∞), and A = [a_{kl}] ∈ M(d, d). Then ∥S|(A, W)∥_{p} := inf c, such that

(9)  ∥ ∑_{k=1}^{d} ∑_{l=1}^{d} a_{kl} ∫_{0}^{T} SU_{t}^{k}dW_{t}^{l}∥_{L_{Y}^{p}} ≤ c∥ ∑_{k=1}^{d} ∫_{0}^{T} U_{t}^{k}dW_{t}^{k}∥_{L_{X}^{p}}

for all T ≥ 0 and (F_{t})_{t≥0}-adapted left-continuous processes of Radon random variables U_{t}^{k} : Ω → X which have right-hand side limits, take values in a finite-dimensional subspace of X, satisfy

∫_{0}^{T} 𝔼∥U_{t}^{k}∥_{X}^{2}dt < ∞

for all T ≥ 0 and k = 1, ..., d, and such that the right hand side of (9) is finite. To shorten the notation we also use in the sequel ∫_{0}^{T} (SU_{t}^{k})_{k=1}^{d} · d(AW)_{t} and ∫_{0}^{T} (U_{t}^{k})_{k=1}^{d} · dW_{t}, respectively, for the expressions inside the norms.

Some minor notation. Given A, B ≥ 0 and c > 0, the notation A ∼_{c} B stands for A/c ≤ B ≤ cA. If the dependence of c on the extra quantities involved is clear, we sometimes simply write A ∼ B.

3. The main result

It will be convenient to have a special notation for particular matrices and multipliers. Thus, we denote

(10) A_{s} := [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}], A_{s,d} := [\begin{matrix}1 & 0 & 0 & … & 0 \\ 0 & −1 & 0 & … & 0 \\ 0 & 0 & −1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & 0 & −1 \end{matrix}], A_{as} := [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}].

We shall also define the special multiplier m_{0} on R^{d}, where d ≥ 2, by

m_{0}(ξ) := 2\frac{ξ_{1}^{2}}{|ξ|^{2}} − 1, equivalently T_{m_{0}} := −(Id + 2R_{1}^{2}).

We recall once more that, for simplicity, all Banach spaces are assumed to be real. Our main result is the following:

Theorem 3.1. Assume that m ∈ C^{∞}(R^{d} \ {0}), d ≥ 2, is a real, homogeneous, and even multiplier that is not identically constant, and A ∈ M(d, d) is a real symmetric matrix that is not a multiple of the identity matrix. Let p ∈ (1, ∞). Then there is a constant C = C(m, A) such that for every pair of Banach spaces X and Y and for every operator S ∈ L(X, Y) one has that

∥S|(A_{s}, W)∥_{p} = UMD_{p}(S) = ∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥

∼_{C} ∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ∼_{C} ∥S|(A, W)∥_{p}.

Exploiting UMD_{p}(R) = p − 1 for p ∈ [2, ∞) (see [9], [11, Theorem 14]) and using that, for d = 2, the multiplier m_{0} corresponds to R_{2}^{2} − R_{1}^{2} and 2R_{1}R_{2} can be obtained by a rotation of m_{0}, Theorem 3.1 implies

Corollary 3.2. For d = 2, p ∈ (1, ∞), and S ∈ L(X, Y) one has

∥Re(BA) ⊗ S : L_{X}^{p}(R^{2}) → L_{Y}^{p} (R^{2})∥

= ∥Im(BA) ⊗ S : L_{X}^{p}(R^{2}) → L_{Y}^{p} (R^{2})∥ = UMD_{p}(S).

In particular, for p ∈ [2, ∞),

∥Re(BA) : L^{p}(R^{2}) → L^{p}(R^{2})∥ = ∥Im(BA) : L^{p}(R^{2}) → L^{p}(R^{2})∥ = p − 1.

We break the proof of Theorem 3.1 into a series of auxiliary results, some of which are basically known, and some of which may have independent interest. The actual proof of Theorem 3.1 is given at the end of this section. Our first step is to show that non-trivial even multipliers dominate the UMD-constant linearly. To this end we need a generalization of a Lemma due to Bourgain [5, Lemma 1]. For the rest of this section we assume that X and Y are Banach spaces and S ∈ L(X, Y) is a fixed operator.

Lemma 3.3. Let p ∈ (1, ∞), Q := T^{d}, and assume that the multiplier m satisfies Assumption 2.1. For k ≥ 1 let E_{k} be the closure in L_{X}^{p}(Q^{k}) of the finite real trigonometric polynomials

Φ_{k}(θ_{1}, ..., θ_{k}) = ∑_{p=1}^{P} Φ_{k}^{p}(θ_{1}, ..., θ_{k})x_{p}

with x_{k} ∈ X and

Φ_{k}^{p} = ∑_{ℓ_{1}∈Z^{d}} ⋯ ∑_{ℓ_{k}∈Z^{d}} e^{i⟨ℓ_{1},θ_{1}⟩} ⋯ e^{i⟨ℓ_{k},θ_{k}⟩}α_{ℓ_{1},...,ℓ_{k}}^{p} where Im(Φ_{k}^{p}) ≡ 0

where only finitely many of the α_{ℓ_{1},...,ℓ_{k}}^{p} ∈ C are non-zero and α_{ℓ_{1},...,ℓ_{k}}^{p} = 0 whenever ℓ_{k} = 0 (so that ∫_{Q} Φ_{k}^{p}(θ_{1}, ..., θ_{k})dθ_{k} = 0). Let T k ; m̃ : L^{p}(Q^{k}) → L^{p}(Q^{k}) be given by

(T k ; m̃Φ_{k})(θ_{1}, ..., θ_{k})

= ∑_{p=1}^{P} ( ∑_{ℓ_{1}∈Z^{d}} ⋯ ∑_{ℓ_{k}∈Z^{d}} m(ℓ_{k})e^{i⟨ℓ_{1},θ_{1}⟩} ⋯ e^{i⟨ℓ_{k},θ_{k}⟩}α_{ℓ_{1},...,ℓ_{k}}^{p} ) x_{p}.

for ℓ_{1}, ..., ℓ_{k} ∈ Z^{d}. Then one has that

∥∑_{k=1}^{n}((T k ; m̃ ⊗ S)Φ_{k})(θ_{1}, …, θ_{k})∥_{L_{Y}^{p} (Q^{n})}

≤ ∥Tm̃ ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥ ∥∑_{k=1}^{n} Φ_{k}(θ_{1}, …, θ_{k})∥_{L_{X}^{p}(Q^{n})}

for Φ_{1} ∈ E_{1},..., Φ_{n} ∈ E_{n}.

Proof. It is sufficient to prove the inequality for finite real trigonometric polynomials Φ_{1}(θ_{1}), ..., Φ_{n}(θ_{1}, ..., θ_{n}). Let A ≥ 1 be an integer, η ∈ Q be an auxiliary variable, denote by T_{m̃,η} the application of Tm̃ with respect to the variable η, and consider the difference

D_{k}^{A}(θ_{1}, ..., θ_{k}, η) := ((T_{m̃,η} ⊗ S)Φ_{k}(θ_{1} + A·, …, θ_{k} + A^{k}·))(η)

−((T k ; m̃ ⊗ S)Φ_{k})(θ_{1} + Aη, …, θ_{k} + A^{k}η).

Note, that in the first term on the right-hand side we apply the multiplier Tm̃ to a function, where θ_{1}, ..., θ_{k}, A act as parameters, whereas in the second term we apply the multiplier T k ; m̃ to Φ_{k} itself. If we show that for our fixed trigonometric polynomial Φ_{k} there is the bound

(11)  sup_{A≥1} A∥D_{k}^{A}∥_{L_{Y}^{∞}} < ∞,

then the proof is completed by, firstly, observing that

∥∑_{k=1}^{n}((T_{m̃,η} ⊗ S)Φ_{k}(θ_{1} + A·, …, θ_{k} + A^{k}·))(η)∥_{L_{Y}^{p} (Q,dη)}

≤ ∥Tm̃ ⊗ S : L_{X}^{p}(Q) → L_{Y}^{p} (Q)∥ ∥∑_{k=1}^{n} Φ_{k}(θ_{1} + Aη, …, θ_{k} + A^{k}η)∥_{L_{X}^{p}(Q,dη)} .

Secondly, one replaces the left-hand side by

∥∑_{k=1}^{n}((T k ; m̃ ⊗ S)Φ_{k})(θ_{1} + Aη, …, θ_{k} + A^{k}η)∥_{L_{Y}^{p} (Q,dη)}

with the corresponding correction terms in A, integrates with respect to the θ’s, applies Fubini’s theorem so that the variable η is removed, and sends A to infinity.

In order to prove (11) we observe that A∥D_{k}^{A}(θ_{1}, ..., θ_{k}, η)∥_{Y} is upper bounded by a finite number of terms of form

A|T_{m̃,η} (e^{i⟨ℓ_{1},θ_{1}+Aη⟩} ⋯ e^{i⟨ℓ_{k},θ_{k}+A^{k}η⟩})

− m(ℓ_{k})e^{i⟨ℓ_{1},θ_{1}+Aη⟩} ⋯ e^{i⟨ℓ_{k},θ_{k}+A^{k}η⟩}||α_{ℓ_{1},...,ℓ_{k}}^{p}|∥Sx_{p}∥_{Y}

with ℓ_{k} ≠ 0. Finally we observe that

A|T_{m̃,η} (e^{i⟨ℓ_{1},θ_{1}+Aη⟩} ⋯ e^{i⟨ℓ_{k},θ_{k}+A^{k}η⟩})

−m(ℓ_{k})e^{i⟨ℓ_{1},θ_{1}+Aη⟩} ⋯ e^{i⟨ℓ_{k},θ_{k}+A^{k}η⟩}|
= A|m(ℓ_{1}A + … + ℓ_{k}A^{k}) − m(ℓ_{k})|
= A|m(ℓ_{1}A^{−k+1} + … + ℓ_{k−1}A^{−1} + ℓ_{k}) − m(ℓ_{k})|

which is bounded in A since m is differentiable at ℓ_{k} ≠ 0. This yields (11).

Proposition 3.4. Assume that m ∈ C^{∞}(R^{d} \ {0}), d ≥ 2, is a smooth, homogeneous, even, and non-constant multiplier. Let δ^{+} := max_{|ξ|=1} m(ξ) and δ^{−} := min_{|ξ|=1} m(ξ) so that δ^{+} − δ^{−} > 0. Then

UMD_{p}(S) ≤ \frac{2}{δ^{+} − δ^{−}} (1 + \frac{|δ^{+} + δ^{−}|}{|δ^{+}| + |δ^{−}|}) ∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥.

In particular, in the case max_{|ξ|=1} m(ξ) = − min_{|ξ|=1} m(ξ) = 1 we have that

UMD_{p}(S) ≤ ∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥.

Proof. By continuity and compactness there are ξ^{−}, ξ^{+} ∈ R^{d} of length one such that m(ξ^{−}) = δ^{−} and m(ξ^{+}) = δ^{+}. Without loss of generality we may assume that ξ^{−} = e_{1} and ξ^{+} = e_{2}, where e_{1} and e_{2} are the first two unit vectors in R^{d} (otherwise (7) enables us to replace m(ξ) by m(Aξ) with suitably chosen A). Define the functions a^{−}, a^{+} ∈ L^{∞}(T^{d}) by a^{−}(θ) := sgn (θ_{1}) and a^{+}(θ) := sgn (θ_{2}) for θ ∈ T^{d} so that Tm̃a^{−} = δ^{−}a^{−} and Tm̃a^{+} = δ^{+}a^{+}. For independent Bernoulli random variables ε_{1}, ε_{2}, … we consider the X-valued Walsh-Paley martingale difference sequence

(ε_{k}d_{k}(ε_{1}, …, ε_{k−1}))_{k=1}^{n}

and a sequence (α_{k})_{k=1}^{n} with α_{k} ∈ {δ^{−}, δ^{+}}. Define ψ_{k} := a^{−} if α_{k} = δ^{−} and ψ_{k} := a^{+} if α_{k} = δ^{+}, and let

ϕ_{k}(θ_{1}, …, θ_{k−1}) := d_{k}(ψ_{1}(θ_{1}), …, ψ_{k−1}(θ_{k−1})).

Since (ψ_{1}(θ_{1}), …, ψ_{n}(θ_{n})) and (ε_{1}, …, ε_{n}) have the same distribution (if we normalize the measure on Q^{n}) and Tm̃ψ_{k} = α_{k}ψ_{k}, Lemma 3.3 implies that

∥∑_{k=1}^{n} α_{k}ε_{k}S_{k}d_{k}(ε_{1}, ..., ε_{k−1})∥_{L_{Y}^{p}}

≤ ∥Tm̃ ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥ ∥∑_{k=1}^{n} ε_{k}d_{k}(ε_{1}, ..., ε_{k−1})∥_{L_{X}^{p}} .

Let A := 2/(δ^{+}−δ^{−}) and B := (δ^{+}+δ^{−})/(δ^{+}−δ^{−}) so that the new sequence β_{k} := Aα_{k} − B satisfies β_{k} = −1 if α_{k} = δ^{−} and β_{k} = 1 if α_{k} = δ^{+}. Then

∥∑_{k=1}^{n} β_{k}ε_{k}S_{k}d_{k}(ε_{1}, ..., ε_{k−1})∥_{L_{Y}^{p}}

≤ [A∥Tm̃ ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥ + |B|∥S∥] ∥∑_{k=1}^{n} ε_{k}d_{k}(ε_{1}, ..., ε_{k−1})∥_{L_{X}^{p}}.

Because ∥m∥_{∞}∥S∥ = sup_{ℓ∈Z^{d}} |m(ℓ)|∥S∥ ≤ ∥Tm̃ ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥ we end up with

UMD_{p}(S) ≤ [A + \frac{|B|}{∥m∥_{∞}} ] ∥Tm̃ ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥
= [A + \frac{|B|}{∥m∥_{∞}} ] ∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥
≤ \frac{2}{δ^{+} − δ^{−}} (1 + \frac{|δ^{+} + δ^{−}|}{|δ^{+}| + |δ^{−}|}) ∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥

where the equality follows from Lemma 2.2.

Proposition 3.5. For T_{m_{0}} = −Id − 2R_{1}^{2}, p ∈ (1, ∞), and d ≥ 2 one has

∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ≤ ∥S|(A_{s,d}, W)∥_{p}. (12)

Proof. We apply the representation of products of Riesz-transforms in terms of heat extensions to the upper half space (see Lemma 6.1). Let f = ∑_{k=1}^{m} f_{k}x_{k} and g = ∑_{l=1}^{n} g_{l}b_{l} with f_{k}, g_{l} ∈ C_{0}^{∞}(R^{d}) and x_{k} ∈ X, b_{l} ∈ Y′. Assume that u_{k} and v_{l} are the heat extensions of f_{k} and g_{l}, respectively, to the upper half plane and that (W_{t})_{t≥0} is a standard Brownian motion in R^{d} starting in the origin. Let u := ∑_{k=1}^{m} u_{k}x_{k}, v := ∑_{l=1}^{n} v_{l}b_{l}, and 1 = (1/p) + (1/p′). Lemma 6.1 gives that

|∫_{R^{d}}⟨((T_{m_{0}} ⊗ S)f)(x), g(x)⟩ dx|

= lim_{T →∞}(2πT)^{d/2}| ∑_{k,l}⟨Sx_{k}, b_{l}⟩ ×

×E (∫_{0}^{T} ∇u_{k}(W_{t}, T − t) · d(A_{s,d}W)_{t} ∫_{0}^{T} ∇v_{l}(W_{t}, T − t) · dW_{t}) |

= lim_{T →∞}(2πT)^{d/2}

|E ⟨∫_{0}^{T} ∇Su(W_{t}, T − t) · d(A_{s,d}W)_{t}, ∫_{0}^{T} ∇v(W_{t}, T − t) · dW_{t}⟩ |.

We continue with

|E ⟨∫_{0}^{T} ∇Su(W_{t}, T − t) · d(A_{s,d}W)_{t}, ∫_{0}^{T} ∇v(W_{t}, T − t) · dW_{t}⟩ |

≤ (E ∥∫_{0}^{T} ∇Su(W_{t}, T − t) · d(A_{s,d}W)_{t}∥_{Y}^{p} )^{1/p}

× (E ∥∫_{0}^{T} ∇v(W_{t}, T − t) · dW_{t}∥_{Y′}^{p′})^{1/p′}

≤ ∥S|(A_{s,d}, W)∥_{p} (E ∥∫_{0}^{T} ∇u(W_{t}, T − t) · dW_{t}∥_{Y}^{p} )^{1/p}

× (E ∥∫_{0}^{T} ∇v(W_{t}, T − t) · dW_{t}∥_{Y′}^{p′})^{1/p′}
= ∥S|(A_{s,d}, W)∥_{p}∥f(W_{T}) − u(0, T)∥_{L_{Y}^{p}} ∥g(W_{T}) − v(0, T)∥_{L_{Y′}^{p′}}

by Itô’s formula because (1/2)Δu_{k} = (∂/∂t)u_{k} and (1/2)Δv_{l} = (∂/∂t)v_{l}. Next, sup_{T ≥0} ∥T^{d/2}u(0, T)∥_{X} < ∞ gives lim_{T →∞} T^{d/2}∥u(0, T)∥_{X}^{p} = 0 so that lim_{T →∞}(2πT)^{d/2}E ∥f(W_{T})−u(0, T)∥_{X}^{p} = ∥f∥_{L_{X}^{p}(R^{d})}^{p}. The same applies for g(W_{T}) and we end up with

|∫_{R^{d}}⟨((T_{m_{0}} ⊗ S)f)(x), g(x)⟩ dx| ≤ ∥S|(A_{s,d}, W)∥_{p}∥f∥_{L_{X}^{p}(R^{d})}∥g∥_{L_{Y′}^{p′}(R^{d})}.

The proof is complete because f and g as above are dense in L_{X}^{p}(R^{d}) and L_{Y′}^{p′}(R^{d}), respectively.

In order to exploit the quantities ∥S|(A, W)∥_{p} in rigorous arguments, one needs at some places approximations of stochastic integrals by discrete martingales. So for the general reader’s convenience we switch to the simple discretized version ∥S|(A, g)∥_{p} introduced below. Just to prove part of our main result one could proceed more directly (see Remark 3.10). Given p ∈ (1, ∞), S ∈ L(X, Y), and A ∈ M(d, d), we let

∥S|(A, g)∥_{p} := inf c,

where the infimum is taken over all c > 0 such that

∥ ∑_{k=1}^{N} ∑_{l=1}^{d} [Sd_{(k−1)d+l}(φ_{1}, ..., φ_{(k−1)d})] ⟨A(φ_{(k−1)d+1}, ..., φ_{kd}), e_{l}⟩∥_{L_{Y}^{p}}

≤ c∥ ∑_{k=1}^{N} ∑_{l=1}^{d} [d_{(k−1)d+l}(φ_{1}, ..., φ_{(k−1)d})] φ_{(k−1)d+l}∥_{L_{X}^{p}} ,

where N = 1, 2, ..., d_{j} : R^{⌊^{\frac{j−1}{d}}⌋d} → X are continuous bounded functions taking values in a finite dimensional subspace of X, and φ_{1}, φ_{2}, ... are independent standard Gaussian random variables. The Nd terms on the right-hand side (in their natural order) we call a Gaussian block martingale difference sequence of order d ≥ 1, on the left-hand side we have its A-transform. Observe that these transforms are not the traditional martingale transforms appearing in the definition of the UMD-spaces.

Lemma 3.6. Let A = [a_{lk}] ∈ M(d, d), S ∈ L(X, Y), and p ∈ (1, ∞). Then the following is true:

(i) ∥S|(A, g)∥_{p} = ∥S|(A, W)∥_{p}. (ii) If U ∈ M(d, d) is real and unitary, then

∥S|(U^{T} AU, g)∥_{p} = ∥S|(A, g)∥_{p}.

(iii) If M ≥ 1 is an integer and the tensor product ⊗^{M}A is defined as the block diagonal matrix with A as each diagonal block, then

∥S|(⊗^{M}A, g)∥_{p} = ∥S|(A, g)∥_{p}.

(iv) Assume a sub-matrix B of A obtained from A by choosing indices I = {k_{1}, …, k_{d′}} with 1 ≤ k_{1} < k_{2} < … < k_{d′} ≤ d and 1 ≤ d′ < d, and deleting the corresponding rows and columns from A. Then

∥S|(B, g)∥_{p} ≤ ∥S|(A, g)∥_{p}.

Proof. (i) It is evident that ∥S|(A, g)∥_{p} ≤ ∥S|(A, W)∥_{p}. The inequality to the other direction follows by a standard approximation of Ito-integrals by discrete Gaussian martingales.

(ii) Here we observe that

∥ ∑_{k=1}^{N} ∑_{l=1}^{d} [Sd_{(k−1)d+l}(φ_{1}, ..., φ_{(k−1)d})] ⟨U^{T} AU(φ_{(k−1)d+1}, ..., φ_{kd}), e_{l}⟩∥_{L_{Y}^{p}}

= ∥ ∑_{k=1}^{N} ⟨U((Sd_{(k−1)d+l}(φ_{1}, ..., φ_{(k−1)d}))_{l=1}^{d}), AU(φ_{(k−1)d+1}, ..., φ_{kd})⟩∥_{L_{Y}^{p}}
≤ ∥S|(A, g)∥_{p} ·

∥ ∑_{k=1}^{N} ⟨U((d_{(k−1)d+l}(φ_{1}, ..., φ_{(k−1)d}))_{l=1}^{d}), U(φ_{(k−1)d+1}, ..., φ_{kd})⟩∥_{L_{Y}^{p}}

= ∥S|(A, g)∥_{p}∥ ∑_{k=1}^{N} ∑_{l=1}^{d} [d_{(k−1)d+l}(φ_{1}, ..., φ_{(k−1)d})] φ_{(k−1)d+l} ∥_{L_{X}^{p}} .

Above we used in the second step the observation that the transformed sequence

⟨U(φ_{1}, ..., φ_{d}), e_{1}⟩, …, ⟨U(φ_{1}, ..., φ_{d}), e_{d}⟩,
⟨U(φ_{d+1}, ..., φ_{2d}), e_{1}⟩, …, ⟨U(φ_{d+1}, ..., φ_{2d}), e_{d}⟩, …

consists again of independent Gaussian random variables because U is unitary.

(iii) The ⊗^{M}A-transform of an appropriate Gaussian martingale block difference sequence of order dM is obtained by simply performing the Atransform of the same sequence (which is also of order d). This shows that ∥S|(⊗^{M}A, g)∥_{p} ≤ ∥S|(A, g)∥_{p}. The converse inequality is a special case of (iv) we treat now.

(iv) We first perform a unitary permutation of the coordinates, which is justified by part (ii), so that we may assume that B = (a_{lk})_{1≤l,k≤d′}. Next we consider the unitary map U which maps the unit vector e_{k} to e_{k} if k = 1, ..., d′ and to −e_{k} if k = d′ + 1, ..., d. From (ii) it follows that, for C := (1/2)(A + U^{T} AU), one has ∥S|(C, g)∥_{p} ≤ ∥S|(A, g)∥_{p}. The entries of C satisfy c_{lk} = a_{lk} if l, k ∈ {1, ..., d′} and c_{lk} = 0 if l ∈ {1, ..., d′} and k ∉ {1, ..., d′}. Now the inequality ∥S|(B, g)∥_{p} ≤ ∥S|(C, g)∥_{p} can be proved by an appropriate augmentation of the Gaussian random variables and of the martingale difference sequence that has to be transformed: we add to each block ϕ_{(k−1)d′+1}, …, ϕ_{kd′} of Gaussian random variables d − d′ independent Gaussian random variables to obtain a block size d and add appropriate zero martingale differences to the original martingale difference sequence. Now the transformation with respect to B can be artificially written as a transformation with respect to C.

The previous lemma will be used to prove the following result:

Proposition 3.7. (i) Assume that A ∈ M(d, d) is real and symmetric, and denote by λ_{max} (respectively, λ_{min}) the largest (respectively, the smallest) eigenvalue of A. Let B ∈ M(d′, d′) be another real and symmetric matrix such that each eigenvalue λ of B satisfies λ_{min} ≤ λ ≤ λ_{max}. Then for any S ∈ L(X, Y) and p ∈ (1, ∞), it holds that

∥S|(B, W)∥_{p} ≤ ∥S|(A, W)∥_{p}.

(ii) Assume that A ∈ M(d, d) is real and antisymmetric, and denote by λ_{max} (respectively, λ_{min}) the largest (respectively, the smallest) eigenvalue of iA. Let B ∈ M(d′, d′) be another real and antisymmetric matrix such that each eigenvalue λ of iB satisfies λ_{min} ≤ λ ≤ λ_{max}. Then for any S ∈ L(X, Y) and p ∈ (1, ∞), it holds that

∥S|(B, W)∥_{p} ≤ ∥S|(A, W)∥_{p}.

Proof. (i) By Lemma 3.6 (i) it is enough to show that ∥S|(B, g)∥_{p} ≤ ∥S|(A, g)∥_{p}. Moreover, by applying the spectral theorem for symmetric matrices and part (ii) of the same Lemma we may assume that both A and B are (real and) diagonal. In addition, by Lemma 3.6 (iii) we may replace A by the tensor product ⊗^{d′}A. Observe that the tensor product ⊗^{d′}A is diagonal and at least d′ of its diagonal elements have the value λ_{max}, and the same holds true for λ_{min}. By applying again a unitary permutation for the coordinates and part (iv) of Lemma 3.6 we obtain for any given sequence Λ := (λ_{1}, …, λ_{d′}) with λ_{j} ∈ {λ_{max}, λ_{min}} for all 1 ≤ j ≤ d′ that the diagonal matrix A_{Λ} ∈ M(d′, d′) (with the diagonal Λ) satisfies

∥S|(A_{Λ}, W)∥_{p} ≤ ∥S|(A, W)∥_{p}.

By the assumption on the eigenvalues of the matrix B, we may express B as a convex combination of matrices of the form A_{Λ}. This clearly yields the claim.

(ii) The matrix iA is self-adjoint, so that the eigenvalues are real. A simple examination of the spectral decomposition of iA (observe that λ and −λ are simultaneously eigenvalues for iA) we may write A, after a unitary transformation, in the form

A = B_{1} ⊗ … ⊗ B_{ℓ},

where each B_{k} is of the form B_{k} = c_{k} A_{as} (A_{as} was defined in (10)), with λ_{min} ≤ c_{k} ≤ λ_{max} (in case d is odd the last one, i.e. B_{ℓ}, equals the 1 × 1 zero matrix and λ_{min} ≤ 0 ≤ λ_{max}). Now the claim follows by an extreme point argument like in (i).

Proposition 3.8. Let m be a homogeneous, even, and smooth multiplier on R^{d}, d ≥ 2. Then for any S ∈ L(X, Y) it holds that

∥T_{m} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ≤ c ∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥

for S ∈ L(X, Y) and p ∈ (1, ∞), where c > 0 depends at most on m.

Proof. In the following we always assume that ξ, θ ∈ S^{d−1}, so that (for example) m_{0}(ξ) = 2ξ_{1}^{2} − 1. Let a ∈ (0, 1). By composing m_{0} with the linear map ξ ↦ B_{a}(ξ) := (\sqrt{1 − a^{2}}ξ_{1}, ξ_{2}, …, ξ_{d}) we infer by (7) that

∥T_{m_{a}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ≤ \frac{1}{1 + a}∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥,

where

m_{a}(ξ) := \frac{2 − a^{2}}{2(1 + a)} + \frac{a^{2}}{2(1 + a)}m_{0} ∘ B_{a}(ξ) = (1 − a)\frac{1}{1 − a^{2}ξ_{1}^{2}}

= (1 − a) ∑_{k=0}^{∞} a^{2k}ξ_{1}^{2k}

and we also use the estimate

∥Id ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ≤ ∥S∥ ≤ ∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥

(for example, use Lemma 2.2 and ∥m_{0}∥_{∞} = 1). Of course, given θ ∈ S^{d−1} we obtain the same estimates for

m_{a}^{θ}(ξ) := (1 − a) ∑_{k=0}^{∞} a^{2k}⟨ξ, θ⟩^{2k}.

Given r ∈ (0, 1), the d-dimensional Poisson kernel for B(0, 1) has the form

P (rξ, θ) = \frac{1}{ω_{d−1}} \frac{(1 − r^{2})}{(1 + r^{2} − 2r⟨ξ, θ⟩)^{d/2}}

= \frac{(1 − r^{2})(1 + r^{2})^{−d/2}}{ω_{d−1}} (1 − \frac{2r⟨ξ, θ⟩}{1 + r^{2}})^{−d/2}

with ω_{d−1} = |∂B(0, 1)|. By the Taylor expansion of y ↦ (1 − y)^{−d/2} we obtain that

P^{s}(rξ, θ) := \frac{1}{2}(P (rξ, θ) + P (rξ, −θ))
= \frac{(1 − r^{2})(1 + r^{2})^{−d/2}}{ω_{d−1}} ∑_{k=0}^{∞} (\frac{2r}{1 + r^{2}} )^{2k} ⟨ξ, θ⟩^{2k}\binom{−d/2}{2k}.

Fix ε > 0 and let

c_{0} := 2ω_{d−1} \frac{Γ(ε + 1)Γ(d/2)}{Γ((d/2) + ε)}.

Using the substitution u = 2r/(1 + r^{2}) and Euler’s β-integral yields that

∫_{0}^{1}(1 + r^{2})^{−2+d/2} (1 − \frac{2r}{1 + r^{2}} )^{d/2−1+ε} P^{s}(rξ, θ) dr

= \frac{1}{2ω_{d−1}} ∑_{k=0}^{∞} (∫_{0}^{1}(1 − u)^{d/2−1+ε}u^{2k} du) \binom{−d/2}{2k}⟨ξ, θ⟩^{2k}

= \frac{1}{2ω_{d−1}} \frac{Γ(d/2 + ε)}{Γ(d/2)} ∑_{k=0}^{∞} (\frac{Γ(d/2 + 2k)}{Γ(d/2 + 2k + 1 + ε)}) ⟨ξ, θ⟩^{2k}

(13)  = \frac{1}{c_{0}} ∫_{0}^{1}(1 − a)^{ε−1}a^{d/2−1}m_{a}^{θ}(ξ) da.

Let us denote by sh(d) a complete orthonormal L^{2}-basis of L^{2}(S^{d−1}, λ), where λ is the normalized Haar measure, consisting of spherical harmonics on S^{d} (we refer to [23, 24] for more details on spherical harmonics). For ψ ∈ sh(d) we denote by deg(ψ) the degree of ψ. Because the function (r, ξ) ↦ r^{k}ψ(ξ) is harmonic if deg(ψ) = k we have in this case

(14)  ∫_{S^{d−1}} P (rξ, θ)ψ(θ) dθ = r^{k}ψ(ξ).

Define

λ_{k} := c_{0} ∫_{0}^{1}(1 + r^{2})^{−2+d/2} (1 − \frac{2r}{1 + r^{2}} )^{d/2−1+ε} r^{k} dr ∼_{c(ε,d)} k^{1−d−2ε}.

Let a_{ψ} := ⟨m, ψ⟩/λ_{k} if ψ ∈ sh(d) and deg(ψ) = k. Applying [23, p. 70] gives

∑_{ψ∈sh_{d}} |a_{ψ}|^{2} < ∞.

Because m is even we find an even f ∈ L^{2}(S^{d−1}, λ) with ⟨f, ψ⟩ = a_{ψ}, so that

(15)  m = ∑_{k=0}^{∞} λ_{k} ∑_{ψ∈sh_{d}, deg(ψ)=k}⟨f, ψ⟩ψ

in L^{2}(S^{d−1}, λ). If we can show that

(16)  m(ξ) = ∫_{S^{d−1}} ∫_{0}^{1}(1 − a)^{ε−1}a^{\frac{d}{2} −1}m_{a}^{θ}(ξ)f(θ)dadθ

for ξ ∈ S^{d−1}, then this would give our assertion since

∫_{S^{d−1}} ∫_{0}^{1}(1 − a)^{ε−1}a^{\frac{d}{2} −1}|f(θ)|dadθ < ∞

implies that m is a convex combination of the multipliers m_{a}^{θ}. In order to verify equality (16) it is sufficient to show that

∫_{S^{d−1}} [∫_{S^{d−1}} ∫_{0}^{1}(1 − a)^{ε−1}a^{\frac{d}{2} −1}m_{a}^{θ}(ξ)f(θ)dadθ] \overline{ψ}(ξ)dξ

= c_{0} ∫_{0}^{1}(1 + r^{2})^{−2+d/2} (1 − \frac{2r}{1 + r^{2}} )^{d/2−1+ε} r^{k} dr⟨f, ψ⟩

for ψ ∈ sh_{d} with deg(ψ) = k. But this follows by a computation from (13) and (14).

Remark 3.9. A slight additional argument shows that one may allow any even multiplier m in the above proposition such that m_{|S^{d−1}} ∈ W^{s,1}(S^{d−1}) for some s > d − 1. Especially, if d = 2 this class contains all m such that for some ε > 0 the derivative D^{ε}m_{|S^{1}} is a function of bounded variation.

We are ready for

Proof of Theorem 3.1. We first verify that UMD_{p}(S) is comparable to the corresponding norm of the operator T_{m_{0}} ⊗ S.

Observe that max_{|ξ|=1} m_{0}(ξ) = 1 and min_{|ξ|=1} m_{0}(ξ) = −1. An application of Proposition 3.4 yields that

UMD_{p}(S) ≤ ∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥. (17)

By Proposition 3.5 we have in turn

∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ≤ ∥S|(A_{s,d}, W)∥_{p}. (18)

The eigenvalues of A_{s,d} and A_{s} are ±1. Hence Proposition 3.7 (i) and Lemma 3.6 (i) yield that

∥S|(A_{s,d}, W)∥_{p} ≤ ∥S|(A_{s}, W)∥_{p} = ∥S|(A_{s}, g)∥_{p}. (19)

However, the A_{s}-transform of discrete Gaussian martingales is a special case of a UMD-martingale transform and hence in case S = Id it follows immediately that

∥S|(A_{s}, g)∥_{p} ≤ UMD_{p}(S). (20)

There are various ways to check this inequality for general S (note that we started with Walsh-Paley martingales in the definition of UMD_{p}(S)). An easy self-consistent way would be to apply the central limit theorem argument from Lemma 4.2 to the Bernoulli variables to replace the Bernoulli variables by the Gaussian random variables (there are arguments to switch from Walsh-Paley martingale difference sequences to arbitrary martingale difference sequences; see for example [10, p. 12] and [20]). By combining the inequalities (17)–(20) we obtain that

UMD_{p}(S) = ∥T_{m_{0}} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ = ∥S|(A_{s}, W)∥_{p}. (21)

This proves the equalities in the theorem.

Now let us assume that A ∈ M(d, d) is symmetric and non-trivial, and that m is an even, smooth, and non-trivial multiplier. Proposition 3.4 shows that UMD_{p}(S) is dominated by a multiple of the norm of T_{m} ⊗ S, and Proposition 3.8 verifies that this norm in turn is dominated by a multiple of the norm of T_{m_{0}} ⊗ S. The linear equivalence of the norm of T_{m} ⊗ S to the UMD-constant of X follows now from (21).

Finally, the equivalence of UMD_{p}(S) to ∥S|(A, W)∥_{p} will be deduced from (21) by an application of Proposition 3.7 (i). Denote by λ_{min}, λ_{max} the minimal and maximal eigenvalue of A. Firstly, we obtain that

∥S|(A, W)∥_{p} ≤ c∥S|(A_{s}, W)∥_{p}

with c := max(|λ_{max}|, |λ_{min}|). To consider the other direction, let α := 2/(λ_{max}−λ_{min}) and β := −(λ_{max}+λ_{min})/(λ_{max}−λ_{min}), so that B := αA+β satisfies λ_{max}(B) = −λ_{min}(B) = 1. Consequently,

∥S|(A_{s}, W)∥_{p} ≤ α∥S|(A, W)∥_{p} + β∥S|(Id, W)∥_{p}
≤ α∥S|(A, W)∥_{p} + \frac{β}{max\{|λ_{max}|, |λ_{min}|\}}∥S|(A, W)∥_{p}

and the proof of Theorem 3.1 is complete.

Remark 3.10. There is a shorter argument for part of Theorem 3.1 which, modulo some ’hand waving’, bypasses Lemma 3.6 and Proposition 3.7. Namely, in order to prove e.g. the statement

UMD_{p}(X) = ∥T_{m_{0}} : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ∼_{C} ∥T_{m} : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥

one only needs Propositions 3.4, 3.5, and 3.8 and ∥S|(A_{s,d}, W)∥_{p} ≤ UMD_{p}(S). Because A_{s,d} is a diagonal matrix, the latter inequality can be suitably approximated by a UMD-transform inequality.

4. Odd multipliers

A canonical example of an odd multiplier is the Hilbert transform. In this section we verify that all odd multipliers, as well as all antisymmetric transforms for stochastic integrals are linearly comparable to the Hilbert transform. Observe that an odd multiplier maps real functions to purely imaginary ones, whence we consider below the operator T_{im} in order to be able to allow also real Banach spaces in the statement.

Theorem 4.1. Assume that m ∈ C^{∞}(R^{d}\{0}), d ≥ 2, is a real non-zero, homogeneous, and odd multiplier, and that A ∈ M(d, d) is a non-zero and anti-symmetric matrix. Let p ∈ (1, ∞). Then there is a constant C = C(A, m) such that for every pair of Banach spaces X and Y and for every operator S ∈ L(X, Y) one has

∥T_{im} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ ∼_{C} ∥S|(A, W)∥_{p} ∼_{C} ∥S|(A_{as}, W)∥_{p}
= ∥ H ⊗S : L_{X}^{p}(R) → L_{Y}^{p} (R)∥.

We start the proof by

Lemma 4.2. For p ∈ (1, ∞) and S ∈ L(X, Y) one has that

∥S|(A_{as}, W)∥_{p} ≤ ∥H̃ ⊗ S : L_{X}^{p}(T) → L_{Y}^{p} (T)∥.

Proof. Assume that M ≥ 1 and that a_{k}, b_{k} : [−1, 1]^{2k−2} → X, k = 1, ..., M, are bounded and continuous. Lemma 3.3 yields that

(22)  ∥ ∑_{k=1}^{M} (H̃ sin(θ_{k})Sa_{k}(sin(θ_{1}), ..., sin(θ_{k−1}); cos(θ_{1}), ..., cos(θ_{k−1}))

+H̃ cos(θ_{k})Sb_{k}(sin(θ_{1}), ..., sin(θ_{k−1}); cos(θ_{1}), ..., cos(θ_{k−1}))∥_{L_{p}^{Y}}

≤ ∥H̃ ⊗ S : L_{p}^{X}(T) → L_{p}^{Y} (T)∥ ×

×∥ ∑_{k=1}^{M} ( sin(θ_{k})a_{k}(sin(θ_{1}), ..., sin(θ_{k−1}); cos(θ_{1}), ..., cos(θ_{k−1}))

+ cos(θ_{k})b_{k}(sin(θ_{1}), ..., sin(θ_{k−1}); cos(θ_{1}), ..., cos(θ_{k−1}))∥_{L_{p}^{X}} .

Next, we apply a blocking argument. Let N, L ≥ 1 be integers and assume that A_{k}, B_{k} : [−1, 1]^{2k−2} → X, k = 1, ..., N, are bounded and continuous functions. For M = NL we apply (22) to

a_{(k−1)L+j}(sin(θ_{1}), ..., sin(θ_{(k−1)L+j−1}); cos(θ_{1}), ..., cos(θ_{(k−1)L+j−1}))

= A_{k}((L/2)^{−1/2}(sin(θ_{1}) + … + sin(θ_{L})),
(L/2)^{−1/2}(sin(θ_{L+1}) + … + sin(θ_{2L})),
…, (L/2)^{−1/2}(sin(θ_{(k−2)L+1}) + … + sin(θ_{(k−1)L}));

(L/2)^{−1/2}(cos(θ_{1}) + … + cos(θ_{L})), (23)

(L/2)^{−1/2}(cos(θ_{L+1}) + … + cos(θ_{2L})),
…, (L/2)^{−1/2}(cos(θ_{(k−2)L+1}) + … + cos(θ_{(k−1)L})))

and with an analogous choice for the coefficients b_{(k−1)L+j} for 1 ≤ k ≤ N and 1 ≤ j ≤ L. By taking into account the action of H̃ on the trigonometric polynomials we obtain that

∥ ∑_{k=1}^{N} ( − Sc_{k}A_{k}(s_{1}, ..., s_{k−1}; c_{1}, ..., c_{k−1}) +

+ Ss_{k}B_{k}(s_{1}, ..., s_{k−1}; c_{1}, ..., c_{k−1}))∥_{L_{p}^{Y}}

≤ ∥H̃ ⊗ S : L_{p}^{X}(T) → L_{p}^{Y} (T)∥ ×

×∥ ∑_{k=1}^{N} (s_{k}A_{k}(s_{1}, ..., s_{k−1}; c_{1}, ..., c_{k−1}) +

+ c_{k}B_{k}(s_{1}, ..., s_{k−1}; c_{1}, ..., c_{k−1}))∥_{L_{p}^{X}}

where we have used the abbreviations s_{k} := (L/2)^{−1/2} ∑_{j=1}^{L} sin(t_{(k−1)L+j}) and c_{k} := (L/2)^{−1/2} ∑_{j=1}^{L} cos(t_{(k−1)L+j}) for k = 1, …, N. By keeping N fixed, letting L → ∞, and by applying the central limit theorem (here we may normalize the Lebesgue measure on T) and the fact that cos and sin are uncorrelated, one gets that

∥S|(A_{as}, g)∥_{p} ≤ ∥H̃ ⊗ S : L_{X}^{p}(T) → L_{Y}^{p} (T)∥.

Finally, Lemma 3.6 verifies that ∥S|(A_{as}, g)∥_{p} = ∥S|(A_{as}, W)∥_{p} and we are done.

The following lemma is well-known. For the convenience of the reader we recall the idea of its proof.

Lemma 4.3. For p ∈ (1, ∞) one has that

∥ H ⊗S : L_{X}^{p}(R) → L_{Y}^{p} (R)∥ ≤ ∥S|(A_{as}, W)∥_{p}.

Proof. We use ∥ H ⊗S : L_{X}^{p}(R) → L_{Y}^{p} (R)∥ = ∥H̃ ⊗ S : L_{X,0}^{p}(T) → L_{Y}^{p} (T)∥ from Lemma 2.2 and consider f(θ) := ∑_{k=1}^{n} (sin(kθ)x_{k} + cos(kθ)y_{k}) with x_{k}, y_{k} ∈ X as a function on the unit circle. Then, a.s.,

u(W_{τ}) = ∫_{0}^{τ} ∇u(W_{t}) · dW_{t} and v(W_{τ}) = − ∫_{0}^{τ} ∇u(W_{t}) · d(A_{as} W)_{t}

by Itô’s formula, where u and v are the harmonic extensions of f and ∑_{k=1}^{n} (− cos(kθ)x_{k} + sin(kθ)y_{k}) to the unit disc, (W_{t})_{t≥0} is a standard twodimensional standard Brownian motion, and τ := inf{t ≥ 0 : |W_{t}| = 1}.

Proof of Theorem 4.1. The equality

∥S|(A_{as}, W)∥_{p} = ∥ H ⊗S : L_{X}^{p}(R) → L_{Y}^{p} (R)∥

follows from Lemmas 4.2, 4.3, and 2.2. Moreover,

∥S|(A, W)∥_{p} ∼_{C} ∥S|(A_{as}, W)∥_{p}

is a consequence of Proposition 3.7 (ii) (note, that the eigenvalues of i A_{as} are ±1 and that iA has at least two symmetric non-zero real eigenvalues). Moreover, by a classical argument called ’the method of rotations’ (see e.g. [15, p.271, formula (4.2.20)] one may express any odd (and smooth) multiplier operator T_{im} as an average of directional Hilbert-transforms which immediately yields that the norm of T_{im} ⊗ S is linearly dominated by that of H ⊗S. Finally, if m is an odd and non-zero multiplier, we may assume that m(e_{1}) = 1. The corresponding discrete multiplier m̃ satisfies −im̃(k_{1}e_{1}) = −isgn (k_{1}) for k_{1} ≠ 0 and −im̃(0e_{1}) = 0. By the consideration of f̃(θ_{1}, ..., θ_{d}) = f(θ_{1}) := ∑_{k=1}^{n} (sin(k_{1}θ_{1})x_{k_{1}} + cos(k_{1}θ_{1})y_{k_{1}}) with x_{k_{1}}, y_{k_{1}} ∈ X and observing that ((H̃ ⊗ S)f)(θ_{1}) = ((−T_{im̃} ⊗ S)f̃)(θ_{1}, ..., θ_{d}) we immediately get that

∥H̃ ⊗ S : L_{X,0}^{p}(T) → L_{Y}^{p} (T)∥ ≤ ∥T_{im̃} ⊗ S : L_{X}^{p}(T^{d}) → L_{Y}^{p} (T^{d})∥

so that ∥H ⊗ S : L_{X}^{p}(R) → L_{Y}^{p} (R)∥ ≤ ∥T_{im} ⊗ S : L_{X}^{p}(R^{d}) → L_{Y}^{p} (R^{d})∥ by Lemma 2.2.

5. Additional remarks

The remaining main open problem is whether or not we have the linear equivalence of the norms

(24)  UMD_{p}(S) ∼ ∥ H ⊗S : L_{X}^{p}(R) → L_{Y}^{p} (R)∥.

By the results obtained in Theorems 3.1 and 4.1 this problem can be formulated now in various ways, for example in a purely probabilistic way via martingale transforms or via multipliers. To find counterexamples to (24) in the category of Banach spaces (i.e. S = Id_{X} for a Banach space X) seems to be harder than to find counterexamples for operators S ∈ L(X, Y), which is one reason for our usage of the operator setting. At the same time the operator setting gives an easier control whether or not estimates are linear (for example UMD_{p}(Id_{X}) < ∞ if and only if ∥ H ⊗Id_{X} : L_{X}^{p}(R) → L_{X}^{p}(R)∥ < ∞ because of Id_{X}^{2} = Id_{X} which does not work for operators). Natural candidates to disprove (24) in the setting of operators are the operators of summation σ_{n} : ℓ_{n}^{1} → ℓ_{n}^{∞}. For them it is known that

∥ H ⊗σ_{n} : L_{ℓ_{n}^{1}}^{2}(R) → L_{ℓ_{n}^{∞}}^{2}(R)∥ ∼ log(n + 1),

see [13] (Section IV, Satz 2.1 (proof) and Korollar 2.4) (cf. also [22] (2.4.4)). So far, the best estimates for the UMD-constants are

\frac{1}{c}\sqrt{log(n + 1)} ≤ UMD_{2}(σ_{n}) ≤ c log(n + 1)

where the lower estimate follows from [14] and the upper one is a consequence of UMD_{2}(Id_{ℓ_{n}^{∞}}) ≤ c log(n + 1) which is folklore.

A determination of the UMD-constant of σ_{n} would be of interest for several reasons: in case of UMD_{2}(σ_{n}) ∼ log(n + 1) this would imply that each non-superreflexive Banach space X contains n-dimensional subspaces E_{n} such that the lower bound UMD_{2}(Id_{E_{n}}) ≥ log(n + 1)/c holds, because due to R.C. James a Banach space X is non-superreflexive if and only if the operators σ_{n} can be uniformly factorized through certain n-dimensional subspaces E_{n} ⊆ X (see [18]). Having the lower estimate one might ask for more connections between the quantitative behavior of the UMD-constants of the finite dimensional subspaces of a Banach space and the property that the space is non-superreflexive. On the other hand, any estimate of type UMD_{2}(σ_{n}) = o(log(n + 1)) would offer some new inside into martingale transforms.

6. Appendix

We recall the well-known connection between singular integrals and Atransforms of stochastic integrals corresponding to heat extensions of functions to the upper half space. A simple proof will be sketched below for the readers convenience. We refer to [2] for the original result, and to [4] and the references therein for corresponding results that use the harmonic extension instead. In order to state the relation let d ≥ 1 and assume a real matrix A = [a_{kl}] ∈ M(d, d). We define the operator U_{A} for smooth elements f, g ∈ C_{0}^{∞}(R^{d}) through the bilinear form

∫_{R^{d}}(U_{A}f)(x)g(x) dx (25)

:= lim_{T →∞}(2πT)^{d/2}E [(∫_{0}^{T} ∇u(W_{t}, T − t) · d(AW)_{t}) ×
× (∫_{0}^{T} ∇v(W_{t}, T − t) · dW_{t})] ,

where u and v are the heat extensions of f and g, respectively, to the upper half space R^{d} × R_{+} (i.e. u(t, x) := 𝔼f(x + W_{t}) and similarly for v), ∇ denotes the differentiation with respect to the x-variables, and (W_{t})_{t≥0} is a standard d-dimensional Brownian motion starting at the origin. Hence, for example, u_{t} = \frac{1}{2}Δu and u(x, 0) = f(x) for x ∈ R^{d}.

Lemma 6.1. Let p ∈ (1, ∞) and d ≥ 2. The operator U_{A} is well-defined and extends to a bounded operator on L^{p}(R^{d}) which can be expressed in terms of the Riesz transforms as

U_{A} = − ∑_{k,l=1}^{d} a_{kl}R_{k}R_{l}.

Proof. We let f, g ∈ C_{0}^{∞}(R^{d}). From Itô’s isometry we obtain that

(2πT)^{d/2}E [(∫_{0}^{T} ∇u(W_{t}, T − t) · d(AW)_{t}) ×

× (∫_{0}^{T} ∇v(W_{t}, T − t) · dW_{t})]

= (2πT)^{d/2}E ∫_{[0,T]}⟨A^{T} ∇u(W_{t}, T − t), ∇v(W_{t}, T − t)⟩dt

= ∫_{R^{d}×[0,T]}⟨A^{T} ∇u(x, t), ∇v(x, t)⟩(2πT)^{d/2}dμ_{t,T} (x)dt,

where μ_{t,T} = law(W_{T −t}). Because of sup_{x∈R^{d},t>0} t^{\frac{d+1}{2}} |∇u(x, t)| < ∞ and similarly for v and by splitting the integration over R^{d} × [0, T] into R^{d} × [0, T/2] and R^{d} × (T/2, T] we see by standard arguments that

lim_{T →∞}(2πT)^{d/2}E [(∫_{0}^{T} ∇u(W_{t}, T − t) · d(AW)_{t}) ×

× (∫_{0}^{T} ∇v(W_{t}, T − t) · dW_{t})] = ∫_{R^{d}×[0,∞)}⟨A^{T} ∇u(x, t), ∇v(x, t)⟩dxdt

(note that ∫_{R^{d}×[0,∞)} |∇u(x, t)|^{2}dxdt < ∞ and the same for v, which follows by the argument below, cf. also [21, Lemma 1.1]). If F u is the Fourier transform of u with respect to the x-variables, then it is well-known that F u(ξ, t) = (F f)(ξ)e^{−t|ξ|^{2}/2} for ξ ∈ R^{d} and t > 0. Observe also that F ((d/dx_{k})u) = iξ_{k}F u for k = 1, ..., d. By Parseval’s formula and Fubini’s theorem we may compute

∫_{R^{d}}(U_{A}f)(x)g(x) dx

= (2π)^{−d} ∫_{R^{d}} (∫_{R^{+}} e^{−t|ξ|^{2}} dt) F f(ξ)\overline{F g(ξ)}⟨iξ, A(iξ)⟩ dξ

= (2π)^{−d} ∫_{R^{d}} F f(ξ)\overline{F g(ξ)}|ξ|^{−2}⟨ξ, Aξ⟩ dξ

= ∫_{R^{d}}(T_{m}f)(x)g(x) dx,

where m is the multiplier m(ξ) := |ξ|^{−2}⟨ξ, Aξ⟩. By recalling that R_{j} corresponds to the multiplier ξ_{j}/(i|ξ|) the claim follows immediately.

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University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FIN-40014 University of Jyväskylä, Finland

E-mail address: geiss@maths.jyu.fi

University of Missouri, Mathematics Department, Columbia, MO 65211 USA

E-mail address: stephen@math.missouri.edu

University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland

E-mail address: eero.saksman@helsinki.fi