6 1. INTRODUCTION
(1.28)
ηpx, y, zq
ÿ
pα1 ,α2,α3
qPA3
xpα1, α2qypα2, α3qzpα1, α3q, px, y, zq P
l2pA2qˆl2pA2qˆl2pA2q.
To obtain a generic integral representation of η, with no restrictions on integrands,
we take the bounded linear map U2 from
l2pA2q
into
L2pΩA, 2
PAq
2
given by
(1.29) U2pxq
ÿ
pα1 ,α2 qPA2
xpα1,α2qrα1 b rα2 , x P
l2pA2q,
and observe
ηpx, y, zq
ż
pω1
,ω2,ω3qPΩA3
U2pxqpω1,ω2qU2pyqpω2,ω3qU2pzqpω1,ω3q dω1dω2dω3,
x P
l2pA2q,
y P
l2pA2q,
z P
l2pA2q.
(1.30)
Adding the requirement that integrands be uniformly bounded, we have
ηpx, y, zq
ż
pω1 ,ω2,ω3
qPΩA3
Φp2,2qpxqpω1, ω2qΦp2,2qpyqpω2, ω3qΦp2,2qpzqpω1, ω3q dω1dω2dω3,
x P
l2pA2q,
y P
l2pA2q,
z P
l2pA2q,
so that
(1.31) Φp2,2q :
l2pA2q
Ñ
L8pΩA, 2 PAq2
is
`
l2pA2q Ñ L2pΩA, 2 PAq 2
˘
- continuous, and
(1.32) }Φp2,2qpxq}L8 ď
K2}x}2,
x P
l2pA2q,
where K ą 1 is the absolute constant in (1.27). The map Φp2,2q is obtained by a
two-fold iteration of Lemma 12.1 in the base case n 2. (See Remark 12.6.i.)
An n-linear version (n ě 3) of the trilinear η in (1.28) is given by
ηnpx1, x2,..., xnq“
ÿ
pα1 ,α2,...,αnqPAn
x1pα1,α2qx2pα2,α3q ¨ ¨ ¨ xnpαn,α1q,
px1, x2,..., xnq P
l2pA2q
ˆ
l2pAq2
ˆ ¨ ¨ ¨ ˆ
l2pA2q,
(1.33)
whose subsequent integral representation is given by
ηnpx1, x2, . . . , xnq

ż
pω1 ,ω2,...,ωn
qPΩn
A
Φp2,2qpx1qpω1, ω2qΦp2,2qpx2qpω2, ω3q ¨ ¨ ¨ Φp2,2qpxnqpωn, ω1q dω1dω2 ¨ ¨ ¨ dωn,
px1 , x2, . . . , xnq P
l2pA2q
ˆ
l2pAq
ˆ ¨ ¨ ¨ ˆ
l2pA2q.
(1.34)
The n-linear functional in (1.33) together with its integral representation in (1.34)
play key roles in the Banach algebra of Hilbert-Schmidt operators. (See Remark
12.9.)
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