Abstract

The classical Grothendieck inequality is viewed as a statement about represen-

tations of functions of two variables over discrete domains by integrals of two-fold

products of functions of one variable. An analogous statement is proved, concerning

continuous functions of two variables over general topological domains. The main

result is the construction of a continuous map Φ from

l2pAq

into

L2pΩA,

PAq, where

A is a set, ΩA “ t´1,

1uA,

and PA is the uniform probability measure on ΩA,

such that

(1)

ÿ

αPA

xpαqypαq “

ż

ΩA

ΦpxqΦpyqdPA, x P

l2pAq,

y P

l2pAq,

and

(2) }Φpxq}L8 ď K}x}2, x P

l2pAq,

for an absolute constant K ą 1. (Φ is non-linear, and does not commute with

complex conjugation.) The Parseval-like formula in (1) is obtained by iterating the

usual Parseval formula in a framework of harmonic analysis on dyadic groups. A

modified construction implies a similar integral representation of the dual action

between lp and lq,

1

p

`

1

q

“ 1.

Variants of the Grothendieck inequality in higher dimensions are derived. These

variants involve representations of functions of n variables in terms of functions of

k variables, 0 ă k ă n. Multilinear Parseval-like formulas are obtained, extending

the bilinear formula in (1). The resulting formulas imply multilinear extensions of

the Grothendieck inequality, and are used to characterize the feasibility of integral

representations of multilinear functionals on a Hilbert space.

Received by the editor November 30, 2011, and, in revised form, November 5, 2012.

Article electronically published on March 11, 2014.

DOI: http://dx.doi.org/10.1090/memo/1093

2010 Mathematics Subject Classification. Primary 46C05, 46E30, Secondary 47A30, 42C10.

Key words and phrases. The Grothendieck inequality, Parseval-like formulas, integral repre-

sentations, fractional Cartesian products, projective continuity, projective boundedness.

Aﬃliation at time of publication: Department of Mathematics, University of Connecticut,

Storrs, Connecticut 06269, email: blei@math.uconn.edu.

c

2014 American Mathematical Society

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