Vector Valued Fourier Transforms and Fourier Type - CiteSeerX

Vector Valued Fourier Transforms and Fourier Type Aicke Hinrichs January 15, 2003

Abstract Since its invention harmonic analysis plays a prominent rˆole in the investigation of real and complex functions. It is desirable to have the same or similar methods available also for vector valued functions. A direct transfer of classical theorems to the vector valued case is, in general, only possible if one considers functions with values in spaces with additional structural properties. This opens up a fascinating interaction between the geometry of the spaces and harmonic analysis. A central question is whether, given a certain Banach space X and a locally compact abelian group G, the X-valued Fourier transform on G still satisfies a Hausdorff-Young inequality. Banach spaces having this property (for a certain p ∈ (1, 2]) are said to be of Fourier type p with respect to G. We will outline the theory around this property with particular emphasis on the connections between the structure of the groups and spaces on the one hand and properties of the Fourier transform on the other hand.

1

Acknowledgement These notes are the lecture notes of a series of talks given at the University of Sevilla in the Functional Analysis Seminar in October 2002. It is my pleasure to express gratitude to the people at the Department of Mathematics for the invitation and the kind hospitality I experienced during my stay there.

1

Introduction

It is a general theme in Functional analysis during the last decades, and in Banach space theory in particular, to investigate if classical results about scalar valued functions remain valid if the functions considered take values in some Banach space. Usually, one of three things happens. The most desirable case having in mind applications is if the results simply remain true in the vector valued setting for any Banach space. The worst case occurs if only “trivial” extensions remain true, possibly for functions with values in finite dimensional spaces or Hilbert spaces. The third and frequently observed case is that it depends on the structure and geometry of the Banach spaces considered whether a result can be carried over to the vector valued setting. The latter case often leads to a fruitful interplay between the geometry of Banach spaces and techniques from the classical theory. Prominent examples are the theory of type and cotype of Banach spaces initiated by Maurey and Pisier [29] and the theory of UMD spaces developed by Burkholder [7, 8, 9] and Bourgain [5]. In these notes we want to look at another example where the extension of scalar valued results to the vector valued setting depends on the structure of the spaces involved - Fourier transforms on locally compact abelian (lca) groups, in particular Hausdorff-Young inequalities for vector valued functions. Here a third component enters into the picture, the structure of the

2

underlying lca group. For the classical Fourier transform on R the study of vector valued Hausdorff-Young inequalities was initiated by Peetre in [32]. His starting point were Banach spaces X satisfying for a given p ∈ [1, 2] that every X-valued function f in the Bochner-Lebesgue space LX p (R) has a 0 Fourier transform in LX p0 (R). Here p is the conjugate number of p given by

1/p+1/p0 = 1. Such spaces are called spaces of Fourier type p. Peetre mainly studied this notion in connection with interpolation methods in [32, 33] as later did Milman in [31] in the case of Fourier transforms on general lca groups. That the Fourier type of a space is mainly tied to its local structure was already apparent in the work of V´agi [38] who proved that a Banach space with an unconditional basis has Fourier type 2 only if the space is isomorphic to a Hilbert space. The unconditionality assumption was soon removed by Kwapie´ n in [25]. His paper played a decisive rˆole in the development of type and cotype theory, see [30]. A systematic study of Hausdorff-Young inequalities for vector valued Fourier transforms on general lca groups was initiated in the work of Andersson [1, 2], Garcia-Cuerva, Kazarian, Kolyada, Torrea [13], and Pietsch, Wenzel [35]. This shall also be the theme in these notes. We want to emphasize here that we do not attempt to give a thorough survey of all results known in this direction. For this, the reader is referred to the just cited sources. Rather we want to give a short introduction into the subject, explain the basic known results, point out some open questions, and provide more details on some of the very recent results. In particular, we neither treat applications of classical Fourier type to the study of abstract Cauchy problems, see [15, 16, 17, 18] nor do we study Fourier type with respect to non-abelian compact groups, see [14].

3

2

Fourier type with respect to lca groups

We will work in the framework of an lca group G which comes equipped with its Haar measure µG . For basic facts on abstract harmonic analysis on lca groups we refer the reader to [12, 19]. Our prime examples are (i) the real line G = R with multiplication, usual topology and Lebesgue measure. (ii) the integers G = Z with addition, discrete topology and counting measure. (iii) the torus G = T of complex numbers of modulus 1 with multiplication, usual topology and normalized Lebesgue measure. We shall often identify T with the interval [0, 1) with addition modulo 1. (iv) the Cantor group D = {−1, 1}N with coordinatewise multiplication, product topology and product measure. A character γ on G is a continuous homomorphism from G into the torus T. The collection of characters on G is an abelian group under pointwise multiplication and carries a natural locally compact topology. The resulting lca group is the dual group G0 of G. In particular, in our examples (i) . . . (iv) the dual groups can be identified with R, T, Z, and the group of Walsh functions, respectively. We use standard Banach space notation as can be found in [27, 28]. For information on vector valued integration and vector valued Lp -spaces we refer to [10]. X and Y will always be Banach spaces and T a linear and bounded operator from X to Y . The dual operator of T is T 0 . For p ∈ [1, ∞], Lp (G) stands for the Lebesgue space Lp (G, µ) and LX p (G) is the Bochner-Lebesgue space of p-integrable X-valued functions. The norm of a function f ∈ LX p (G) is denoted by kf kLX or short kf kp . For a function f ∈ L1 (G), the Fourier p (G) 4

transform FG f is given by Z (FG f )(γ) =

f (x)γ(x)dµG (x) G

for γ ∈ G0 . It is a function in C0 (G0 ), the space of continuous functions on G0 vanishing at infinity. We also write fˆ for FG f if it is clear from the context which group G is considered. The classical Hausdorff-Young inequality for G makes it possible to extend the Fourier transform also to functions in Lp (G) for 1 < p ≤ 2 so that FG defines a bounded operator from Lp (G) into Lp0 (G0 ). We assume the standard normalization of the Haar measure µG so that Plancherel’s Theorem holds which means that FG is an isometry from L2 (G) onto L2 (G0 ). From now on, p will always be in the interval [1, 2]. The Fourier transform can be easily extended to functions in the algebraic tensor product Lp (G)⊗X P consisting of finite sums f = ϕk xk with ϕk ∈ Lp (G) and xk ∈ X by letting P FG f = (FG ϕk )xk ∈ Lp0 (G) ⊗ X. Since Lp (G) ⊗ X is dense in LX p (G) the question arises whether FG can be extended to a bounded and linear X 0 operator from LX p (G) into Lp0 (G ). That this is not true in general can be

seen by the following simple but instructive example. Example. Let G = Z, G0 = T = [0, 1), X = L∞ (T). For a fixed integer N , let the function fN in Lp (Z) ⊗ X be given as fN =

N X

ek ⊗ e2πik .

k=−N

where (ek ) is the unit vector basis in Lp (Z) = lp (Z). Considering fN as a X function in lX (Z) and fc N as a function in L 0 ([0, 1)), we obtain p

kfN kp =

p

à X k

Ã

!1/p kfN (k)kpX

N X

=

k=−N

5

!1/p ke2πik . kp∞

= (2N + 1)1/p

and µZ

1

kfc N kp0 = 0

p0

ÃZ

¶1/p0

kfc N (s)kX ds

1

= 0

!1/p0 N ° X ° 0 p ° e2πik( . −s) °∞ ds = 2N +1. k=−N

In particular, if p > 1 then there cannot be a constant c independent of N such that kfc N kp0 ≤ ckfN kp , so an extension of the Fourier transform to a 0 X linear and bounded operator from LX p (G) into Lp0 (G ) is not possible.

We now define that the Banach space X has Fourier type p with respect to G if the Fourier transform FG extends to a bounded linear operator from 0 X LX p (G) into Lp0 (G ). This just means that a Hausdorff-Young inequality

kFG f kLX0 (G0 ) ≤ ckf kLX p (G) p

(1)

holds for all X-valued functions f ∈ LX p (G). The operator norm of the extended operator is denoted by kX|FT G p k. This definition is easily extended to the case of operators. We will say that T has Fourier type p with respect to G if the operator FG ⊗ T originally defined from Lp (G) ⊗ X to Lp0 (G0 ) ⊗ Y extends to a bounded linear operator Y 0 from LX p (G) to Lp0 (G ). Again, the operator norm of the extended operator G G is then denoted by kT |FT G p k, so that kIX |FT p k = kX|FT p k where IX

is the identity operator on X. The class of all operators of Fourier type p with respect to G equipped with the norm k · |FT G p k is a Banach operator ideal in the sense of Pietsch, see [34, 35]. We denote this ideal by FT G p. In particular, the ideal property says that if T ∈ FT G p and A, B are linear bounded operators such that AT B is defined then AT B ∈ FT G p . Observe that every operator has Fourier type 1 with respect to any lca group. Interpolation methods can be used to show that the ideals FT G p are linearly ordered with respect to p, i.e. for 1 < p1 < p2 < 2 we have G G G FT G 2 ⊂ FT p2 ⊂ F T p1 ⊂ FT 1 = L

6

where L denotes the class of all linear and bounded operators between Banach spaces. Moreover, if G is infinite, all these inclusions are strict. In particular, G n’s result already mentioned Lp1 ∈ FT G p1 \ FT p2 , see [13]. Moreover, Kwapie´

in the introduction can be extended to show that a Banach space has Fourier type 2 with respect to some infinite lca group if and only if it is isomorphic to a Hilbert space, see [13]. The following question which is still open asks whether this characterization has a counterpart for operators. Here H denotes the class of all operators T factoring through a Hilbert space, i.e. for which there exists a Hilbert space H and linear bounded operators R from X to H and S from H to Y such that T = SR. Question 1. If G is an infinite group, is it true that FT G 2 = H? In other words, does every operator of Fourier type 2 with respect to G factor through Hilbert space? Observe that H ⊂ FT G n’s Theorem and the ideal 2 follows from Kwapie´ property of FT G 2 . Question 1 is even open for the special cases where G is one of our examples (i) . . . (iv). An affirmative answer to this question would mean that the concept of Fourier type 2 does not depend on the underlying lca group G. The analog question for p < 2 is open even for spaces. Question 2. Do the ideals FT G p depend at all on the infinite lca group G? In particular, if G1 and G2 are infinite lca groups and p ∈ (1, 2), does there exist a Banach space X or an operator T which has Fourier type p with respect to G1 but not with respect to G2 ? For special groups G1 and G2 , the question whether Fourier type p can be transferred from the group G1 to the group G2 has gained considerable 1 2 1 2 attention. Transference principles FT G ⊂ FT G or FT G = FT G are p p p p

known in a number of cases. The first results in this direction were concerned with the classical groups R, T, Z. It was independently shown by K¨onig [24], Andersson [1], Garcia-Cuerva, Kazarian, Kolyada, Torrea [13] that d

d

d

FT Rp = FT Zp = FT Tp = FT Rp = FT Zp = FT Tp 7

(2)

for any positive integer d. In [2] Andersson also got some abstract transference principles to subgroups and quotient groups reading as follows. G2 1 • If G2 is an open subgroup of G1 then FT G p ⊂ FT p . G1 /G2 1 • If G2 is a compact subgroup of G1 then FT G . p ⊂ FT p

We conclude this section quoting a fundamental result of Bourgain from [4, 6] which says that B-convex spaces are just the spaces which have a Fourier type p for some p > 1 with respect to the classical groups or the Cantor group. Recall that a Banach space is B-convex if and only if it has some nontrivial Rademacher type if and only if it does not contain the spaces l1n uniformly, see [29].

3

Duality

In this section we want to look at the duality theory of Fourier type in more detail to show how functional analytic methods and methods from abstract harmonic analysis work together. The Pontrjagin duality for lca groups tells us that the bidual group G00 can be naturally identified with the group G. This duality fits together very well with Banach space duality to produce the following standard duality theorem. This result (for Banach spaces) was proved in [2, 13] but was probably folklore before. Theorem 3.1. A bounded linear operator T has Fourier type p with respect to the lca group G if and only if the dual operator T 0 has Fourier type p with 0

G 0 respect to the dual group G0 . Moreover, in this case kT |FT G p k = kT |FT p k

Proof. We first show that 0

G kT 0 |FT G p k ≤ kT |FT p k.

8

(3)

0

To this end let f ∈ LYp (G). By [11, p. 1173], for a given ε > 0 we can find 0 a function g ∈ LX p (G ) such that kgkp = 1 and Z 0ˆ kT f kp0 ≤ (1 + ε) hg(γ), T 0 fˆ(γ)i dµG0 (γ). G0

Using Pontrjagin duality and H¨older’s inequality, we obtain that Z Z 0ˆ hg(γ), T f (γ)i dµG0 (γ) = hT gˆ(s−1 ), f (s)i dµG (s) ≤ kT gˆkp0 kf kp G0

G 0

≤ kT |FT G p k kf kp . Since ε > 0 was arbitrary, we find that kT 0 fˆkp0 ≤ kT |FT G p k kf kp 0

0

for all f ∈ LYp (G) proving (3). Finally, applying (3) twice and taking local reflexivity and Pontrjagin duality into account yields 0

00

G G G G 00 0 kT |FT G p k = kT |FT p k ≤ kT |FT p k ≤ kT |FT p k = kT |FT p k

proving the theorem. Now the examination of our prime examples shows that even more is 0

G true in these cases, in fact we have that FT G p = FT p in the cases G =

R, T, Z, D which follows from (2) in the classical case and is shown for G = D in [13, 35]. The natural question which lca groups exhibit this kind of autoduality behavior was raised in [13]. It was recently answered in [22] where it was shown that this autoduality is actually shared by all lca groups: Theorem 3.2. For any bounded linear operator T between Banach spaces and all lca groups G, the following properties are equivalent: (i) T has Fourier type p with respect to G. (ii) T 0 has Fourier type p with respect to G. 9

(iii) T has Fourier type p with respect to G0 . (iv) T 0 has Fourier type p with respect to G0 . In particular, this theorem also shows that Fourier type p does not only pass to subspaces (as is immediate from the definition) but also to quotient spaces. We reproduce the proof here in some detail to demonstrate how the structure theory of lca groups enters. One main observation is that any known transference principle extends to product groups as follows. Theorem 3.3. Let G1 , G2 and H be lca groups and 1 < p ≤ 2. If there exists a constant c such that G1 2 kT |FT G p k ≤ ckT |FT p k 1 for all operators T ∈ F T G p then also 2 ×H 1 ×H kT |FT G k ≤ ckT |FT G k p p 1 ×H for all operators T ∈ F T G . In particular, (2) implies for all d ∈ N that p d

d

d

FT R×G = FT Z×G = FT T×G = FT pR ×G = FT pZ ×G = FT pT ×G . (4) p p p q 0 Remark. Let Bp = p1/p /p0 1/p be the Babenko-Beckner constant. Then actually the following equalities and inequalities can be shown: d ×H

kT |FT Zp

d ×H

k = kT |FT Z×H k ≤ Bp−1 kT |FT T×H k = kT |FT pT p p

kT |FT Rp

d ×H

k ≤ kT |FT Tp

d ×H

k ≤ Bp−d kT |FT pR

d ×H

k,

k.

Theorem 3.3 follows from the next theorem which connects the Fourier type of an operator T with respect to a product group G×H with the Fourier Y 0 type of the tensor product FH ⊗ T : LX p (H) → Lp0 (H ). It was observed in

[2] that T ∈ FT G×H already implies T ∈ FT H p p , so that the latter operator is well defined. In [2] only spaces are considered, but the extension of the proof to the operator case is straightforward. 10

Theorem 3.4. Let G and H be lca groups and 1 < p ≤ 2. Then T ∈ FT G×H p G X Y 0 if and only if T ∈ F T H p and FH ⊗ T : Lp (H) → Lp0 (H ) ∈ FT p . Moreover,

in this case G×H k. kFH ⊗ T |FT G p k = kT |FT p G X Y 0 Proof. Assume first that T ∈ FT H p and FH ⊗ T : Lp (H) → Lp0 (H ) ∈ FT p .

We will prove that kT |FT G×H k ≤ kFH ⊗ T |FT G p p k =: c.

(5)

This implies that T ∈ FT G×H . By density, to verify (5) it is enough to show p that

°X ° ° ° (FG×H %j )T xj ° ° j

LY (G0 ×H 0 ) p0

°X ° ° ° ≤ c° % j xj ° j

LX p (G×H)

holds for all finite families %j ∈ Lp (G × H) and xj ∈ X. Again by density of Lp (G) ⊗ Lp (H) in Lp (G × H) this is equivalent to the inequality °X ° °X ° ° ° ° ° FG×H (ϕj ψj )T xj ° Y ≤ c° ϕj ψj xj ° ° 0 0 X Lp0 (G ×H )

j

Lp (G×H)

j

for all ϕj ∈ Lp (G), ψj ∈ Lp (H) and xj ∈ X. Since FG×H (ϕj ψj )T xj = (FG ϕj )(FH ψj )T xj = (FG ϕj )(FH ⊗ T )(ψj ⊗ xj ) and

°X ° ° ° ϕ ψ x ° j j j° j

LX p (G×H)

°X ° ° ° =° ϕj (ψj ⊗ xj )° j

LX (H)

Lp p

(G)

,

(6)

(7)

this is immediate from the definition of c = kFH ⊗ T |FT G p k. Now assume that T ∈ FT G×H . By the remark before the statement of p the theorem, we know that T ∈ FT H p . We will prove that G×H kFH ⊗ T |FT G k =: d. p k ≤ kT |FT p

Again density arguments reduce this to the question whether ° ° °X °X ° ° ° ° ϕj ψj xj ° (FG ϕj )(FH ⊗ T )(ψj ⊗ xj )° LY0 (H 0 ) ≤ d° ° p j

Lp0

11

(G0 )

j

(8)

LX (H)

Lp p

(G)

holds for all finite families ϕj ∈ Lp (G), ψj ∈ Lp (H) and xj ∈ X. The proof of (8) is finished by the observation that the definition of d = kT |FT pG×H k together with (6), (7) and °X ° ° ° (FG ϕj )(FH ⊗T )(ψj ⊗xj )° ° j

LY0 (H 0 )

Lp0p

(G0 )

°X ° ° ° =° (FG ϕj )(FH ψj )T xj ° j

LY (G0 ×H 0 ) p0

imply this inequality. Finally, (5) and (8) also show the claimed equality of norms. 1 ×H Proof of Theorem 3.3. If T ∈ F T G , we obtain from Theorem 3.4 that p 1 FH ⊗ T ∈ FT G p and 1 ×H 1 kT |FT G k = kFH ⊗ T |FT G p p k. 2 Now the assumption implies that FH ⊗ T ∈ FT G p and

G1 2 kFH ⊗ T |FT G p k ≤ ckFH ⊗ T |FT p k. 2 ×H Applying Theorem 3.4 once more, we find that T ∈ FT G and p 2 ×H 2 kT |FT G k = kFH ⊗ T |FT G p p k.

Altogether, we proved the claim. We now recall a result of Andersson from [2] which he used for the proof of the transference principle for open subgroups mentioned above. Here Cc (H, X) denotes the space of all compactly supported X-valued functions on the lca group H. Proposition 3.5. Let G be an lca group which contains an open subgroup H. For any f ∈ Cc (H, X), the extension g to all of G defined by zero outside of H satisfies

kFG ⊗ T gkLY0 (G0 ) p

kgkLX p (G)

=

12

kFH ⊗ T f kLY0 (H 0 ) p

kf kLX p (H)

.

Now we can prove our main duality result in the case that G is compact or discrete. 0

Theorem 3.6. Let G be a compact or discrete abelian group. Then FT G p ⊂ FT G p and 0

kT |FTpG k ≤ Bp−1 kT |FTpG k 0

for all T ∈ FT G p . Proof. By the standard duality Theorem 3.1 it is sufficient to consider the case that G is discrete. By a density argument, it is then enough to prove 0

k(FG ⊗ T )gkLY0 (G0 ) ≤ Bp−1 kT |FT G p k kgkLX p (G) p

(9)

for functions g with finite support on G. So suppose that g is such a function with finite support S ⊂ G given by g(a) = xa ∈ X for a ∈ S and g(a) = 0 for a ∈ G \ S. The finitely generated group H = hSi is topologically isomorphic to Zd × F for some d ∈ N ∪ {0} and some finite abelian group F . Then we can consider the restriction f of g to H. By Proposition 3.5, we obtain that (9) is equivalent to 0

k(FH ⊗ T )f kLY0 (H 0 ) ≤ Bp−1 kT |FT G p k kf kLX p (H) p

(10)

To prove this inequality, we first observe that by the definition of Fourier type with respect to H k(FH ⊗ T )f kLY0 (H 0 ) ≤ kT |FT H . p k kf kLX p (H) p

The remark after Theorem 3.3 implies that d ×F

Z kT |FT H p k = kT |FT p

k. k = kT |FT Z×F k ≤ Bp−1 kT |FT T×F p p

Since F is finite the dual group of F is isomorphic to F . Then the standard duality Theorem 3.1 and the remark after Theorem 3.3 give d ×F

kT |FT T×F k = kT 0 |FT Z×F k = kT 0 |FT pZ p p 13

k = kT 0 |FT H p k.

Since G is discrete, H is an open subgroup of G. Applying Proposition 3.5 then implies together with the standard duality Theorem 3.1 0

G G 0 kT 0 |FT H p k ≤ kT |FT p k = kT |FT p k.

Altogether, we obtain (10) which completes the proof of the theorem. The preceding theorem is already sufficient to prove Theorem 3.2 for the special case of compact or discrete abelian groups. The next theorem brings us a step closer to the general case, which will then follow from the structure theorem of lca groups. Theorem 3.7. Let G be an lca group which has a compact and open subgroup. 0

G Then FT G p ⊂ FT p and 0

0

kT |FTpG k ≤ Bp−3 kT |FTpG k for all T ∈ FT G p .

(11)

Proof. Let us first assume that G is topologically isomorphic to Zd × K for some d ∈ N ∪ {0} and some compact abelian group K. Using again the remark after Theorem 3.3 and Theorem 3.6, we get d ×K

Z kT |FT G p k = kT |FT p

k = kT |FT Z×K k ≤ Bp−1 kT |FT T×K k p p 0

0

d ×K 0

≤ Bp−2 kT |FT Z×K k ≤ Bp−3 kT |FT pT×K k = Bp−3 kT |FT pT p

k

G0

= Bp−3 kT |FT p k. Now assume the general case that we just have an open and compact subgroup H of G. Let q : G → G/H be the canonical quotient map and observe that G/H is a discrete abelian group. By a density argument, it is enough to show that 0

k(FG ⊗ T )gkLY0 (G0 ) ≤ Bp−3 kT |FT G p k kgkLX p (G) p

(12)

for functions g vanishing outside of q −1 (S) for some finite set S ⊂ G/H. So let g be such a function and let M = q −1 (hSi) be the preimage of the finitely 14

generated group hSi which is topologically isomorphic to Zd × F for some d ∈ N ∪ {0} and some finite abelian group F . Then M is an open subgroup of G containing H as a subgroup and M/H is topologically isomorphic to Zd × F . Now it follows from the proof of Theorem 9.8 in [19] that M itself is topologically isomorphic to Zd × K for some compact abelian group K, so we already proved (11) for M . Let f be the restriction of g to M . We apply (11) for M to obtain that 0

k(FM ⊗ T )f kLY0 (M 0 ) ≤ kT |FT M ≤ Bp−3 kT |FT M . p k kf kLX p k kf kLX p (M ) p (M ) p

Using the standard duality Theorem 3.1 and Proposition 3.5 we find that 0

0

M G G 0 0 kT |FT M p k = kT |FT p k ≤ kT |FT p k = kT |FT p k.

Hence 0

k(FM ⊗ T )f kLY0 (M 0 ) ≤ kT |FT G . p k kf kLX p (M ) p

Since g vanishes outside of M , (12) follows by Proposition 3.5, which concludes the proof. Now we are prepared for the proof of the complete duality theorem. Proof of Theorem 3.2. The equivalence of (i) and (iv) as well as the equivalence of (ii) and (iii) follow from the standard duality Theorem 3.1. Then the theorem is proved if we show that (iii) implies (i) for again this also gives that (ii) implies (iv) by Theorem 3.1. So it is enough to show for any lca group G and all operators T ∈ FT G p

0

that 0

kT |FTpG k ≤ CkT |FTpG k

(13)

for some constant C depending only on p and G. We already know this from Theorem 3.7 if G contains a compact and open subgroup. The general case will now follow from the structure theorem for lca groups saying that a 15

general lca group is topologically isomorphic to Rd × H for some d ∈ N ∪ {0} and some lca group H which has a compact and open subgroup. We obtain from the remark after Theorem 3.3 and Theorem 3.1 that d ×H

kT |FTpG k = kT |FTpR

d ×H

k ≤ kT |FTpT

k.

Since H contains a compact and open subgroup, the same is true for Td × H. Applying Theorem 3.7, we find that d ×H

kT |FTpT

d ×H 0

k ≤ Bp−3 kT |FTpZ

k.

Finally, the remark after Theorem 3.3 gives that kT |FTpZ

d ×H 0

d ×H 0

k ≤ Bp−d kT |FTpR

0

k = Bp−d kT |FTpG k,

so that (13) indeed holds with C = Bp−d−3 .

4

Fourier type 2 and Hilbert space factorization

In this section we will take a closer look at the case p = 2. Of course, restricted to spaces, Kwapien’s Theorem gives a final answer to the structure of Banach spaces of Fourier type 2 with respect to an infinite lca group G: they are isomorphic to Hilbert spaces. That Hilbert spaces have Fourier type 2 immediately implies that every operator factoring through a Hilbert space also has Fourier type 2. Whether the converse is true is the open Question 1 from the introduction. A recent result [23] gives at least some information: Theorem 4.1. An operator of Fourier type 2 with respect to the classical groups R, T, Z has Fourier type 2 with respect to every lca group G. The special case G = D was already treated in [21], a different proof can be found in [26]. 16

Both notions of Fourier type and factorization through a Hilbert space are local notions in the sense that they depend only on the finite dimensional pieces of the operator under consideration. This is made precise in the following two theorems. Proofs can be found in [35]. Both theorems are mainly due to Kwapie` n [25], see also [24] and [36]. Theorem 4.2. For any operator T the following assertions are equivalent. (i) T has Fourier type 2 with respect to R. (ii) There exists c > 0 such that for all sequences x1 , x2 , . . . in X °2 Z1 ° ∞ ∞ °X ° X ° ° 2 exp(2πikt)T xk ° dt ≤ c kxk k2 . ° ° ° k=1

0

k=1

(iii) There exists c > 0 such that for all n = 1, 2, . . . and x1 , . . . , xn ∈ X ° °2 n °X n n ° X X 1 ° ° √ exp(2πihk/n)T xk ° ≤ c2 kxk k2 . (14) ° ° ° n h=1 k=1 k=1 Theorem 4.3. For any operator T the following assertions are equivalent. (i) T factors through a Hilbert space. (ii) There exists c > 0 such that for all orthonormal systems (a1 , a2 , . . .) in some Hilbert space L2 (M, µ) and all sequences x1 , x2 , . . . in X °2 Z ° ∞ ∞ °X ° X ° ° ak (t)T xk ° dµ(t) ≤ c2 kxk k2 . ° ° ° M

k=1

k=1

(iii) There exists c > 0 such that for all n = 1, 2, . . . and all n × n-matrices An = (αhk ) satisfying kAn : l2n → l2n k ≤ 1 and x1 , . . . , xn ∈ X °2 ° n n n °X ° X X ° ° 2 kxk k2 . αhk T xk ° ≤ c ° ° ° h=1

k=1

k=1

17

(15)

Remark. A similar characterization as in Theorem 4.2 is possible for Fourier type p operators, see [35]. We restricted ourselves to the case p = 2 to make the similarity between both theorems more striking. For a fixed n × n-matrix An , define κ(T |An ) as the smallest constant c such that (15) holds for all x1 , . . . , xn ∈ X. Let also κn (T ) = sup{κ(T |An ) : kAn : l2n → l2n k ≤ 1}. Observe that Theorem 4.3 says that the sequence of ideal norms (κn ) is a natural gradation in the sense of [37] of the Hilbert space factorization norm generally defined as inf kAk kBk where the infimum is taken over all factorizations T = AB over a Hilbert space. Similarly, letting En =

√1 (exp(2πihk/n), n

Theorem 4.2 expresses that (κ(An )) is a natural gradation of the Fourier type 2 norm. Considering the problem whether every Fourier type 2 operator factors through a Hilbert space it is natural to ask whether the sequences of ideal norms κn and κ(En ) are uniformly equivalent, i.e. whether there exists a constant c such that κn (T ) ≤ cκ(T |En ) for all operators T . The following theorem from [20] shows that this is not true. Theorem 4.4. For any sequence (An ) of n × n-matrices which satisfies kAn : l2n → l2n k ≤ 1 there exists a sequence (Tn ) of operators such that κn (Tn ) = 1

and

lim nα κ(Tn |An ) = 0 for α