4 1. INTRODUCTION

where rσx are the coordinate functions given by (1.4), whence

(1.17)

ÿ

α

xpαqypαq “

ż

Ω

φ1pxqφ2pyqdμ, x P

l2,

y P

l2,

with Ω “ ΩB

l2

ˆ ΩB

l2

and μ “ |λ|{}λ}M . Now let

Φ “

φ1 ` φ2

2

` i

φ1 ´ φ2

2

,

which, notably, is neither linear nor continuous.

The Grothendieck inequality proper implies (1.13) with a probability measure

μ on Ω “ ΩB

l2

ˆ ΩB

l2

, and a non-linear map

Φ :

l2

Ñ

L8pΩBl2

ˆ ΩBl2 , μq,

which is not continuous with respect to the ambient topologies of

l2

and

L8pΩ,μq.

A question arises: can we do better? That is, can a Parseval-like formula be derived

with a more wieldy and canonical pΩ, μq, and with a map Φ that somehow ”detects”

the structures of its domain and range?

In the first part (Chapter 2 - Chapter 9), given sets X and Y , we study repre-

sentations of scalar-valued functions of two variables, x P X and y P Y , by integrals

whose integrands are two-fold products of functions of one variable, x P X and

y P Y separately. If X and Y are merely sets, then the Grothendieck inequality –

seen as an integral representation of an inner product – plays a natural role. If X

and Y are topological spaces, then to play that same role, the Grothendieck inequal-

ity needs an upgrade. The main result (Theorem 3.5) is an integral representation

of the dot product

(1.18)

ÿ

αPA

xpαqypαq “

ż

ΩA

ΦpxqΦpyqdPA, x P Bl2pAq, y P Bl2pAq,

where A is an infinite set,

ΩA :“ t´1,

1uA,

PA :“ uniform product measure (normalized Haar measure),

and the map

(1.19) Φ : Bl2pAq Ñ

L8pΩA,

PAq

is uniformly bounded, and also continuous in a prescribed sense.

The proof of Theorem 3.5 is carried out in a setting of harmonic analysis on

dyadic groups. The construction of Φ implicitly uses Λp2q-uniformizability [4], a

property of sparse spectral sets manifested here through the use of Riesz products.

Facts and tools drawn from harmonic analysis are reviewed as we move along (e.g.,

Chapter 4).

In Chapter 9, a modification of the proof of Theorem 3.5 yields a Parseval-like

formula for xx, yy, x P

lp,

y P

lq,

1 ď p ď 2 ď q ď 8,

1

p

`

1

q

“ 1. (Theorem 9.1).

1.4. Multilinear Parseval-like formulas

In the second part (Chapter 10 - Chapter 12), we study representations of

functions of n variables (n ě 2) by integrals whose integrands involve functions of

k variables, k ă n. In this context we consider extensions of the (two-dimensional)

Grothendieck inequality to dimensions greater than two.