Optimal exponents for Hardy--Littlewood inequalities for $m$-linear operators

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by  R. M. Aron

OPTIMAL EXPONENTS FOR HARDY–LITTLEWOOD INEQUALITIES FOR
m-LINEAR OPERATORS
R. M. ARON,D. NU´N˜EZ-ALARCO´N,D.M. PELLEGRINO,AND D.M. SERRANO-RODR´IGUEZ
6
1 Abstract. TheHardy–Littlewood inequalities on ℓp spaces provideoptimal exponentsfor some
0 classes of inequalities for bilinear forms on ℓp spaces. In this paper we investigate in detail the
2 exponentsinvolvedinHardy–Littlewoodtypeinequalitiesandprovideseveraloptimalresultsthat

n werenotachievedbythepreviousapproaches. Ourfirstmainresultassertsthatforq1,...,qm >0
u and an infinite-dimensional Banach space Y attaining its cotypecotY, if
J 1 1 1
+...+ < ,
8 p1 pm cotY
2
thenthe following assertions are equivalent:

(a) There is a constant CY ≥1 such that
] p1,...,pm
A 1
F ∞ ∞ ∞ qmqm−1 qq12 q1
h.   ··· kA(ej1,...,ejm)kqm! ···  ≤CpY1,...,pmkAk
jX1=1 jX2=1 jXm=1
t  
ma for all continuousm−linear operators A:ℓp1×···×ℓpm →Y. 
(b) The exponentsq1,...,qm satisfy
[ q1 ≥λmp1,,c.o..t,pYm,q2 ≥λmp2−,..1.,,cpomtY,...,qm ≥λp1,mcotY,
4 where, for k=1,...,m,
v
cotY
8 λpk,...,pm := .
m−k+1,cotY
7 1− 1 +...+ 1 cotY
1 pk pm
0 As an application of the above result we gener(cid:16)alize to the m(cid:17)-linear setting one of the classical
0 Hardy–Littlewood inequalities for bilinear forms. Our result is sharp in a very strong sense: the
. constants and exponentsare optimal, even if we consider mixed sums.
2
0
6
1
: 1. Introduction
v
Xi Let K be the real or complex scalar field. In 1934 Hardy and Littlewood proved three theorems
(Theorems 1.1, 1.2, 1.3, below) on the summability of bilinear forms on ℓ × ℓ (here, and
r p q
a henceforth, when p = ∞ we consider c instead of ℓ ). For any function f we shall consider
0 ∞
f(∞):= lim f(s) and for any s ≥ 1 we denote the conjugate index of s by s∗, i.e., 1+ 1 = 1.
s→∞ s s∗
For all p,q ∈ (1,∞], such that 1 + 1 < 1, let us define
p q
pq
λ := ,
pq−p−q
and
4pq
µ = .
3pq−2p−2q
2010 Mathematics Subject Classification. 47B37, 47B10.
Key words and phrases. Absolutely summing operators; multilinear forms; multilinear operators; Hardy–
Littlewood inequality.
R. Aron is supported in part by MINECO MTM2014-57838-C2-2-P and Prometeo II/2013/013, D. Nu´n˜ez is
supported by Capes Grant 000785/2015-06, D. Pellegrino is supported by CNPq and D. M. Serrano is supported
byCAPES Grant 000786/2015-02.
1
2 ARON,NU´N˜EZ,PELLEGRINO,ANDSERRANO
If p and q are simultaneously ∞, then λ and µ are 1 and 4/3 respectively.
From now on, (e )∞ denotes the sequence of canonical vectors in ℓ .
k k=1 p
Theorem 1.1. (See Hardy and Littlewood [13, Theorem 1]) Let p,q ∈ [2,∞], with 1 + 1 ≤ 1.
p q 2
There is a constant C ≥ 1 such that
p,q
1
λ λ
∞ ∞ 2
(1) |A(e ,e )|2 ≤ C kAk,
  j1 j2   p,q
jX1=1 jX2=1
 
 
 
and
1
∞ µ
(2) |A(e ,e )|µ ≤ C kAk,
 j1 j2  p,q
j1X,j2=1
 
for all continuous bilinear forms A :ℓ ×ℓ → K.
p q
It is well known that the exponents λ and µ are optimal. Also, in (1) the positions of the
exponents 2 and λ can be interchanged. Furthermore, 2 and λ can be replaced by a,b ∈ [λ,2]
provided that
1 1 3 1 1
+ ≤ − + .
a b 2 p q
(cid:18) (cid:19)
Theorem 1.2. (See Hardy and Littlewood [13, Theorem 2]) Let p,q ∈[2,∞], with 1 < 1+1 < 1.
2 p q
There is a constant C ≥ 1 such that
p,q
1
λ λ
∞ ∞ 2
(3) |A(e ,e )|2 ≤ C kAk,
  j1 j2   p,q
jX1=1 jX2=1
 
 
 
and
1
∞ λ
(4) |A(e ,e )|λ ≤ C kAk,
 j1 j2  p,q
j1X,j2=1
 
for all continuous bilinear forms A :ℓ ×ℓ → K.
p q
The exponent λ above is also optimal. However, contrary to what happens in Theorem 1.1,
now, in (3) the exponents 2 and λ cannot be interchanged (see [10]).
Theorem 1.3. (See Hardy and Littlewood [13, Theorem 3]) Let 1 < q < 2 < p, with 1 + 1 < 1.
p q
There is a constant C ≥ 1 such that
p,q
1
∞ λ
(5) |A(e ,e )|λ ≤ C kAk,
 j1 j2  p,q
j1X,j2=1
 
for all continuous bilinear forms A :ℓ ×ℓ → K.
p q
The “optimal” exponent in (5) was improved in [16]:
OPTIMAL EXPONENTS FOR HARDY–LITTLEWOOD INEQUALITIES FOR m-LINEAR OPERATORS 3
Theorem 1.4. (See Osikiewicz and Tonge [16]) Let 1 < q ≤ 2 < p, with 1 + 1 < 1. If
p q
A: ℓ ×ℓ → K is a continuous bilinear form, then
p q
1
∞ ∞ qλ∗ λ
(6) |A(e ,e )|q∗ ≤ kAk.
  j1 j2  
jX1=1 jX2=1
 
 
 
Hardy–Littlewood type inequalities were extensively investigated in recent years, but despite
much progress there are still several open questions concerning the optimality of exponents and
constants.
One of the main nuances on the optimality of exponents that apparently has been overlooked
in the past is that results of optimality of exponents for expressions like
1
∞ s
|A(e ,...e )|s ≤ CkAk
 j1 jm 
j1,..X.,jm=1
 
are in some sense sub-optimal. The main point is that the above inequality can be viewed as
1
∞ ∞ ∞ s1msm−1 sm1−1 s1
(7)  ...  |A(ej1,...ejm)|sm  ... ≤ CkAk
jX1=1 jmX−1=1 jXm=1 
 
   
 
 
for s = ... = s = s, and this is the way that the optimality of the exponents can be investigated
1 m
with more accuracy. A simple illustration of this fact is that the exponent λ of (4) is optimal, but
a quick look at (3) shows that the optimality of (4) is just apparent. An extensive investigation of
the Hardy–Littlewood inequalities in light of multiple sums like (7) was initiated in [3, 4, 5], but
there are still some subtle issues not encompassed by previous work. One of the main technical
obstacles is to develop methods to find optimal exponents in situations in which the optimal
exponents of each sum cannot be interchanged. This is the case of our first main result (for
definition of cotype, see the next section):
Theorem. (See Theorem 2.2, below) Let q ,...,q > 0 and Y be an infinite-dimensional Banach
1 m
space attaining its cotype cotY. If
1 1 1
+...+ < ,
p p cotY
1 m
then the following assertions are equivalent:
(a) There is a constant CY ≥ 1 such that
p1,...,pm
∞ ∞ ∞ qmqm−1 qq12 q11
  ··· kA(ej1,...,ejm)kqm ···  ≤ CpY1,...,pmkAk
jX1=1 jX2=1 jXm=1 
 
   
 
 
for all continuous m-linear operators A: ℓ ×···×ℓ → Y.
p1 pm
(b) The exponents q ,...,q satisfy
1 m
q ≥λp1,...,pm, q ≥ λp2,...,pm ,..., q ≥ λpm−1,pm, q ≥ λpm ,
1 m,cotY 2 m−1,cotY m−1 2,cotY m 1,cotY
4 ARON,NU´N˜EZ,PELLEGRINO,ANDSERRANO
where, for k = 1,...,m,
cotY
λpk,...,pm := .
m−k+1,cotY
1− 1 +...+ 1 cotY
pk pm
(cid:16) (cid:17)
Despitethewidegenerality oftheresultsof[3,4,12],theresultsofthispaperdonotfollow from
the techniques developed in these earlier papers. We illustrate, by means of a concrete example,
how the above Theorem provides more precise information than previously known results.
Example 1.5. Suppose that m = 3, p = p = p = 10, and Y = ℓ . The above Theorem implies
1 2 3 3
that there is a universal constant C ≥ 1 such that
∞ ∞ ∞ qq23 qq12 q11
(8)    kA(ej1,ej2,ej3)kq3   ≤ CkAk
jX1=1 jX2=1 jX3=1 
 
   
 
 
for all continuous 3-linear forms A: ℓ ×ℓ ×ℓ → ℓ if and only if
10 10 10 3
q ≥ 30,
1


 q2 ≥ 125,


q ≥ 30,
 3 7




while the best previously known estimates(from [12, Proposition 4.3] and [3, Theorem 1.5]) just
give that (8) is valid for q ≥ 30 for all j = 1,2,3 and that we cannot have simultaneously
j
q = q = q < 30.
1 2 3
Our second main result, stated and proved in Section 3, is an application of this Theorem,
generalizing Theorems 1.3 and 1.4 with optimal exponents, to the multilinear setting. In Section
4weshowthattheoptimalconstant forthescalar-valued caseisprecisely 1, andfinallyweremark
how our results can be translated to the theory of multiple summing operators.
2. Optimal exponents: vector-valued case
Let 2 ≤ q < ∞ and 0 < s < ∞. Recall that (see [1]) a Banach space X has cotype q if there is
a constant C > 0 such that, no matter how we select finitely many vectors x ,...,x ∈X,
1 n
1 s 1/s
n q n
(9) kx kq ≤ C r (t)x dt ,
j j j
   (cid:13) (cid:13) 
j=1 Z[0,1](cid:13)j=1 (cid:13)
X (cid:13)X (cid:13)
   (cid:13) (cid:13) 
where r denotes the j-th Rademacher function. (cid:13)It is well kn(cid:13)own that if (9) is satisfied for a
j (cid:13) (cid:13)
certain s > 0, then it is satisfied for all s > 0. For a fixed s, the smallest of these constants will
be denoted by C (X) and the infimum of the cotypes of X is denoted by cotX. By convention
q,s
we denote C (X) by C (X).
q,2 q
The following simple lemma will be useful.
OPTIMAL EXPONENTS FOR HARDY–LITTLEWOOD INEQUALITIES FOR m-LINEAR OPERATORS 5
Lemma 2.1. Let Y be a Banach space, m ≥ 2, p ,...,p ∈ [1,∞], and q ,...,q ,r ,...,r ∈
1 m 1 m 2 m
(0,∞). Assume that if
∞ ∞ ∞ qmqm−1 qq23 q12
  ··· kA(ej2,...,ejm)kqm ···  < ∞
jX2=1 jX3=1 jXm=1 
 
   
 
 
for all continuous (m−1)-linear operators A : ℓ × ··· × ℓ → Y, then q ≥ r for all
p2 pm i i
i∈ {2,...,m}. Then
∞ ∞ ∞ qmqm−1 qq12 q11
  ··· kB(ej1,...,ejm)kqm ···  < ∞
jX1=1 jX2=1 jXm=1 
 
   
 
for all continuous m-linear operators B : ℓ × ··· × ℓ → Y implies that q ≥ r for all
p1 pm i i
i∈ {2,...,m}.
Proof. Let A : ℓ × ··· × ℓ → Y be a continuous (m−1)-linear operator and consider the
p2 pm
continuous m-linear operator B :ℓ ×···×ℓ → Y given by
1 p1 pm
B (x(1),...,x(m)) = x(1)A x(2),...,x(m) .
1 1
(cid:16) (cid:17)
Clearly kB (e ,e ,...,e )k = kA(e ,...,e )k, and since
1 1 j2 jm j2 jm
∞ ∞ ∞ qmqm−1 qq12 q11
  ··· kB1(ej1,...,ejm)kqm ··· 
jX1=1 jX2=1 jXm=1 
 
   
 
 ∞ ∞ qmqm−1 q12 
= ··· kB (e ,e ,...,e )kqm ···
  1 1 j2 jm  
jX2=1 jXm=1
 
 
 ∞ ∞ qmqm−1 q12
= ··· kA(e ,...,e )kqm ··· ,
  j2 jm  
jX2=1 jXm=1
 
 
the proof is done.   (cid:3)
From now on, let r ≥ 2, and let p ,...,p ∈ (r,∞] be such that
1 m
1 1 1
+...+ < .
p p r
1 m
For k = 1,...,m, we define
r
λpk,...,pm := .
m−k+1,r
1− 1 +...+ 1 r
pk pm
For a Banach space Y and 1 ≤ s ≤ ∞, let ℓ ((cid:16)Y) be the Ban(cid:17)ach space of Y−valued sequences
s
(y )∞ with the norm
i i=1
1
∞ s
k(y )∞ k = ky ks ,
i i=1 ℓr(Y) i Y!
i=1
X
6 ARON,NU´N˜EZ,PELLEGRINO,ANDSERRANO
(the usual modification is required if s = ∞). When there is no ambiguity, for a vector y ∈ Y, we
denote the norm kyk by kyk.
Y
We now state and prove our first main theorem. As we mentioned before, it improves [12,
Proposition 4.3] and [3, Theorem 1.5], by providing the exact optimal exponents.
Theorem 2.2. Let q ,...,q > 0, and Y be an infinite-dimensional Banach space with cotype
1 m
cotY. If
1 1 1
+...+ < ,
p p cotY
1 m
then the following assertions are equivalent:
(a) There is a constant CY ≥ 1 such that
p1,...,pm
∞ ∞ ∞ qmqm−1 qq12 q11
  ··· kA(ej1,...,ejm)kqm ···  ≤ CpY1,...,pmkAk
jX1=1 jX2=1 jXm=1 
 
   
 
 
for all continuous m-linear operators A :ℓ ×···×ℓ → Y.
p1 pm
(b) The exponents q ,...,q satisfy
1 m
q ≥ λp1,...,pm,q ≥ λp2,...,pm ,...,q ≥ λpm−1,pm,q ≥ λpm .
1 m,cotY 2 m−1,cotY m−1 2,cotY m 1,cotY
Proof. From now on, we shall denote r = cotY. The proof of the case m = 1 can be verified by
usinga shortargument from thetheory of absolutely summingoperators, butwe preferto present
a self contained argument. It suffices to note that
cotY rp
λp1 = = 1 ,
1,cotY 1− cotY p1−r
p1
and
1 1
n r n r
rp1 r r
kA(ej)kp1−r = A kA(ej)kp1−r ej
   
Xj=1 Xj=1(cid:13) (cid:16) (cid:17)(cid:13)
(cid:13) (cid:13)
 1 n   (cid:13) 2 12 (cid:13) 
r
≤ Cr(Y) rj(t)kA(ej)kp1−r A(ej) dt
 (cid:13) (cid:13) 
Z0 (cid:13)j=1 (cid:13)
(cid:13)X (cid:13)
 (cid:13) (cid:13) 
(cid:13) n (cid:13)
≤ Cr(Y) sup (cid:13) rj(t)kA(ej)kp1r−r A(ej)(cid:13)
(cid:13) (cid:13)
t∈[0,1](cid:13)j=1 (cid:13)
(cid:13)X (cid:13)
(cid:13) n (cid:13)
(cid:13) r (cid:13)
≤ Cr(Y) sup (cid:13) ϕ kA(ej)kp1−r A(ej) (cid:13)
ϕ∈BY∗Xj=1(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) 1 (cid:12) 1∗
≤ Cr(Y) sup n kA(ej)kpr1p−1r p1 n |ϕ(A(ej))|p∗1 p1
   
ϕ∈BY∗ j=1 j=1
X X
 1   
n p1
rp1
≤ Cr(Y) kA(ej)kp1−r kAk.
 
j=1
X
 
OPTIMAL EXPONENTS FOR HARDY–LITTLEWOOD INEQUALITIES FOR m-LINEAR OPERATORS 7
So, if (b) is true, then (a) holds.
Assume (a). By the Maurey-Pisier factorization result (see [15] and [11, pg. 286,287]) the
infinite-dimensional Banach space Y finitely factors the formal inclusion ℓ ֒→ ℓ , i.e., there are
r ∞
constants C ,C > 0 such that for all n there are vectors z ,...,z ∈Y satisfying
1 2 1 n
1/r
n n
C (a )n ≤ a z ≤ C |a |r
1 j j=1 ∞ (cid:13) j j(cid:13) 2 j 
(cid:13) (cid:13) (cid:13)(cid:13)Xj=1 (cid:13)(cid:13) Xj=1
(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)  
for all sequences of scalars (aj)nj=1. Consi(cid:13)(cid:13)der the c(cid:13)(cid:13)ontinuous linear operator An : ℓp1 → Y given
by
n
A (x) = x z .
n j j
j=1
X
Since
1 1 1
+ = ,
p λp1 r
1 1,r
we have, using the H¨older inequality,
n 1
kAnk = sup xjzj ≤ C2nλp1,1r.
(cid:13) (cid:13)
kxk≤1(cid:13)j=1 (cid:13)
(cid:13)X (cid:13)
(cid:13) (cid:13)
On the other hand, there is a constant CpY1 =(cid:13)(cid:13)C, such(cid:13)(cid:13)that
1
n q1
CkAnk ≥ kAn(ej)kq1 ≥ C1nq11.
 
j=1
X
 
Since n is arbitrary, q ≥ λp1 (i.e. (b) holds), and this concludes the proof of the case m = 1.
1 1,r
The proof of the general case is performed by induction on m. We know that the result is valid
for m = 1 and we shall prove that it is valid for a certain m whenever it is valid for m−1.
(a)⇒(b). Let us suppose that
1 1 1
+...+ < .
p p r
1 m
A fortiori,
1 1 1
+...+ <
p p r
2 m
and, by our induction hypothesis, if there is a constant CY ≥ 1 such that
p2,...,pm
∞ ∞ ∞ qmqm−1 qq23 q12
  ··· kA(ej2,...,ejm)kqm ···  ≤ CpY2,...,pmkAk
jX2=1 jX3=1 jXm=1 
 
   
 
 
8 ARON,NU´N˜EZ,PELLEGRINO,ANDSERRANO
forallcontinuous(m−1)-linearoperatorsA: ℓ ×···×ℓ → Y,thenbyLemma2.1weconclude
p2 pm
that (a) implies
q ≥ λp2,...,pm,
2 m−1,r
.
.
.
q ≥ λpm−1,pm,
m−1 2,r
q ≥ λpm.
m 1,r
So, we must only show that
q ≥ λp1,...,pm.
1 m,r
As for the m = 1 case, there are constants C ,C > 0 such that for all n there are vectors
1 2
z ,...,z ∈ Y satisfying
1 n
1/r
n n
(10) C (a )n ≤ a z ≤ C |a |r
1 j j=1 ∞ (cid:13) j j(cid:13) 2 j 
(cid:13) (cid:13) (cid:13)(cid:13)Xj=1 (cid:13)(cid:13) Xj=1
(cid:13) (cid:13)
for all sequences of scalars ((cid:13)a )n .(cid:13)Consid(cid:13)(cid:13)er the co(cid:13)(cid:13)ntinuous multilinear operator A :ℓ ×···×
j j=1 (cid:13) (cid:13) n p1
ℓ → Y given by
pm
n
A (x(1),...,x(m)) = x(1)x(2)...x(m)z .
n j j j j
j=1
X
Since
m
1 1 1
+ = ,
λp1,...,pm p r
m,r k
k=1
X
by the H¨older inequality we obtain
1/r
n n
r
(1) (m) (1) (m)
kA k = sup x ...x z ≤ sup C x ...x
n (cid:13) j j j(cid:13) 2 j j 
kx(1)k,···.kx(m)k≤1(cid:13)(cid:13)Xj=1 (cid:13)(cid:13) kx(1)k,···,kx(m)k≤1 Xj=1(cid:12) (cid:12)
(cid:12) (cid:12)
≤ sup (cid:13)(cid:13)(cid:13)C m n x((cid:13)(cid:13)(cid:13)k) pk 1/pk n |1|λmp1,,r...,pm (cid:12)λmp1,,r.1..,pm (cid:12) 
2  j   
kx(1)k,···,kx(m)k≤1 kY=1 Xj=1(cid:12) (cid:12) Xj=1
 (cid:12) (cid:12) 
1   (cid:12) (cid:12)   
≤ C2nλmp1,,r...,pm.
On the other hand, by (10)
n n n qmqm−1 qq12 q11
  ··· kAn(ej1,...,ejm)kqm ··· 
jX1=1 jX2=1 jXm=1 
 
   
 
 1 1 
n q1 n q1
= kAn(ej,...,ej)kq1 = kzjkq1 ≥ C1nq11,
   
j=1 j=1
X X
   
and, since n is arbitrary,
q ≥ λp1,...,pm.
1 m,r
OPTIMAL EXPONENTS FOR HARDY–LITTLEWOOD INEQUALITIES FOR m-LINEAR OPERATORS 9
(b)⇒(a). Let A : ℓ ×···× ℓ → Y be a continuous m-linear operator and define, for all
p1 pm
positive integers n,
An,e : ℓp1 ×···×ℓpm−1 → ℓλp1,mr (Y)
by
n
A (x(1),...,x(m−1)) = A x(1),...,x(m−1),e .
n,e j
j=1
We assert that (cid:16) (cid:16) (cid:17)(cid:17)
kA k≤ C (Y)kAk.
n,e r
To see this, since Y has cotype r and using the H¨older inequality,
1
n rpm r
A x(1),...,x(m−1),e pm−r
j
 
Xj=1(cid:13) (cid:16) (cid:17)(cid:13)
(cid:13) (cid:13)
 (cid:13) (cid:13)  1
n r r r
= A x(1),...,x(m−1), A x(1),...,x(m−1),e pm−r e
j j
 
Xj=1(cid:13)(cid:13) (cid:18) (cid:13) (cid:16) (cid:17)(cid:13) (cid:19)(cid:13)(cid:13)
(cid:13) (cid:13)
 (cid:13)(cid:13) 1 n (cid:13) (cid:13)r (cid:13)(cid:13)  2 12
≤ C (Y) r (t) A x(1),...,x(m−1),e pm−r A x(1),...,x(m−1),e dt
r j j j
 (cid:13) (cid:13) 
Z0 (cid:13)(cid:13)Xj=1 (cid:13) (cid:16) (cid:17)(cid:13) (cid:16) (cid:17)(cid:13)(cid:13)
(cid:13) (cid:13)
 (cid:13)(cid:13) n (cid:13) (cid:13) r (cid:13)(cid:13) 
≤ C (Y) sup (cid:13) r (t) A x(1),...,x(m−1),e pm−r A x(1),...,x(m−1),e (cid:13)
r j j j
(cid:13) (cid:13)
t∈[0,1](cid:13)(cid:13)Xj=1 (cid:13) (cid:16) (cid:17)(cid:13) (cid:16) (cid:17)(cid:13)(cid:13)
(cid:13) (cid:13)
(cid:13) n (cid:13) (cid:13)r (cid:13)
≤ C (Y) sup (cid:13)(cid:13) ϕ A x(1),...,x(m−1),e pm−r A x(1),...,x(m−1),e (cid:13)(cid:13)
r j j
ϕ∈BY∗Xj=1(cid:12)(cid:12) (cid:18)(cid:13) (cid:16) (cid:17)(cid:13) (cid:16) (cid:17)(cid:19)(cid:12)(cid:12)
(cid:13) (cid:13)
n(cid:12)(cid:12) (cid:13) r(cid:13)pm p1m n (cid:12)(cid:12) p∗ p1∗m
≤ C (Y) sup A x(1),...,x(m−1),e pm−r ϕ A x(1),...,x(m−1),e m
r j j
   
ϕ∈BY∗ Xj=1(cid:13) (cid:16) (cid:17)(cid:13) Xj=1(cid:12) (cid:16) (cid:16) (cid:17)(cid:17)(cid:12)
(cid:13) (cid:13) (cid:12) (cid:12)
 (cid:13) (cid:13) 1   (cid:12) (cid:12) 
n rpm pm
≤ C (Y) A x(1),...,x(m−1),e pm−r A x(1),...,x(m−1),· .
r j
 
Xj=1(cid:13) (cid:16) (cid:17)(cid:13) (cid:13) (cid:16) (cid:17)(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
Therefore,  (cid:13) (cid:13)  (cid:13) (cid:13)
pm−r
n rpm rpm
A x(1),...,x(m−1),e pm−r ≤ C (Y)kAk x(1) ··· x(m−1)
j r
 
Xj=1(cid:13) (cid:16) (cid:17)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
and thus  (cid:13) (cid:13)  (cid:13) (cid:13) (cid:13) (cid:13)
kA k = sup A x(1),...,x(m−1)
n,e n,e
kx(1)k,···,kx(m−1)k≤1
(cid:13) (cid:16) (cid:17)(cid:13)
(cid:13)(cid:13) (cid:13)(cid:13) pm−r
n rpm rpm
= sup A x(1),...,x(m−1),e pm−r
j
 
kx(1)k,···,kx(m−1)k≤1 Xj=1(cid:13) (cid:16) (cid:17)(cid:13)
(cid:13) (cid:13)
≤ C (Y)kAk,  (cid:13) (cid:13) 
r
as required.
10 ARON,NU´N˜EZ,PELLEGRINO,ANDSERRANO
On the other hand, since X = ℓλpm (Y) has cotype λp1,mr := R (because λp1,mr > r = cotY) and
1,r
1 1 1 1 1
+...+ < − = ,
p p cotY p cotX
1 m−1 m
we can use the induction hypothesis (with the (m−1)-linear operator A ), and conclude that if
n,e
q ≥ λp1,...,pm−1,q ≥ λp2,...,pm−1,...,q ≥ λpm−1,
1 m−1,R 2 m−2,R m−1 1,R
then
n n n n qmR−1 qqmm−−12 qq12 q11
  ··· kA(e ,...,e )kR ··· 
  j1 jm  
jX1=1jX2=1 jmX−1=1 jXm=1  
   
       
 
 n n n qqmm−−12 qq21 q11 
=   ··· An,e(ej1,...,ejm−1) Xqm−1 ··· 
jX1=1 jX2=1 jmX−1=1(cid:13) (cid:13) 
 
  (cid:13) (cid:13)  
 
≤ CX kA k 
p1,...,pm−1 n,e
≤ CX C (Y)kAk.
p1,...,pm−1 r
Now, the proof is almost done, since
R
λpk,...,pm−1 =
m−k,R
1−R 1 + 1 +...+ 1
pk pk+1 pm−1
(cid:16) λpm (cid:17)
1,r
=
1−λpm 1 + 1 +...+ 1
1,r pk pk+1 pm−1
(cid:16) rpm (cid:17)
= pm−r
1− rpm 1 + 1 +...+ 1
pm−r pk pk+1 pm−1
= λpk,...,pm (cid:16) (cid:17)
m−k+1,r
for each k ∈ {1,...,m−1}.
To conclude the proof we just need to remark that
∞ ∞ ∞ ∞ qmqm−1 qqmm−−21 qq12 q11
  ··· kA(e ,...,e )kqm ··· 
  j1 jm  
jX1=1jX2=1 jmX−1=1 jXm=1  
   
       
 
 ∞ ∞ ∞ ∞ qmR−1 qqmm−−12 qq21 q11
≤   ··· kA(e ,...,e )kR ··· 
  j1 jm  
jX1=1jX2=1 jmX−1=1 jXm=1  
   
       
 
provided q ≥R = λpm.  (cid:3)
m 1,r