Sharp generalizations of the multilinear Bohnenblust--Hille inequality

SHARP GENERALIZATIONS OF THE MULTILINEAR BOHNENBLUST–HILLE INEQUALITY

arXiv:1306.3362v3 [math.FA] 11 Jul 2013

´ N. ALBUQUERQUE, F. BAYART, D. PELLEGRINO, AND J.B. SEOANE-SEPULVEDA Abstract. We prove that the multilinear Bohnenblust–Hille is a particular case of a quite general family of optimal inequalities.

1. Introduction The Bohnenblust-Hille inequality constant C = C(m) ≥ 1 such that  ∞ X  (1.1)

(see [6]) asserts that for all positive integers m ≥ 2 there is a

i1 ,...,im =1

A(ei1 , ..., eim )

2m m+1

 m+1 2m 

≤ C kAk

for all continuous m-linear forms A : c0 × · · · × c0 → K. Here, as usual, K denotes the fields of real or complex scalars. This inequality has important applications in various fields of analysis and mathematical physics ([3, 14]). The case m = 2 is the well-known Littlewood’s 4/3 inequality [13]. In this paper we show that the Bohnenblust–Hille inequality is a very particular case of a large family of sharp inequalities. More precisely, we prove the following general result: Theorem 1.1. Let m ≥ 1, let q1 , ..., qm ∈ [1, 2]. The following assertions are equivalent: (1) There is a constant Cq1 ...qm ≥ 1 such that 





P ∞ P   ∞    ...  i1 =1 i2 =1

∞ P

im−1 =1

∞ P

|A (ei1 , ..., eim )|qm

im =1

qm−2 qm−1 ! q qm

for all continuous m-linear forms A : c0 × · · · × c0 → K. (2) q11 + · · · + q1m ≤ m+1 2 .

m−1

1  q2  qq12 q1 q3     ···  ≤ Cq1 ...qm kAk   

The Bohnenblust–Hille inequality is just the particular case 2m . q1 = · · · = qm = m+1 This kind of inequalities was already considered in [2] when m = 2, as generalizations of Littlewood’s 4/3 inequality. The strategy for the proof of (2) =⇒ (1) in Theorem 1.1 will be very simple, maybe simpler than all previous known proofs of the Bohnenblust-Hille inequality. The starting point is the generalized Littlewood mixed (ℓ1 , ℓ2 )−norm inequality, that is that Theorem 1.1 is true when (q1 , . . . , qm ) = (1, 2, . . . , 2). This property is well-known and it is a consequence of Khintchine’s inequality. Using this and nothing else than Minkowski’s inequality and interpolation, we will deduce the general case. Several generalizations of the Bohnenblust-Hill inequality have already been obtained. Let us quote two of them. The first one is a vector-valued version due to Defant and Sevilla-Peris in [8]. Key words and phrases. Bohnenblust–Hille inequality; Littlewood’s 4/3 inequality; Absolutely summing operators. 1

2

ALBUQUERQUE, BAYART, PELLEGRINO, AND SEOANE

Theorem - Defant/Sevilla-Peris. Let m ≥ 1, 1 ≤ s ≤ q ≤ 2. Define ρ=

2m . m + 2 1s − q1

Then there exists a constant C > 0 such that, for every continuous m-linear mapping A : c0 × · · · × c0 → ℓs , then 1/ρ  +∞ X  ≤ CkAk. kA(ei1 , . . . , eim )kρq  i1 ,...,im =1

Second, in the spirit of the Hardy-Littlewood generalization of Littlewood’s 4/3 inequality, Praciano-Pereira has studied in [17] the effect of replacing c0 by ℓp in the Bohnenblust-Hille inequality. For the sake of convenience, let us introduce the following notations which will be used throughout the paper: for p = (p1 , . . . , pm ) ∈ [1, +∞]m , let 1 = 1 + ···+ 1 . p p1 pm For p ≥ 1, let also Xp = ℓp and let us define X∞ = c0 .

Theorem - Praciano-Pereira. Let m ≥ 1, 1 ≤ p1 , . . . , pm ≤ +∞ with p1 ≤ 12 . Define ρ=

2m

. m + 1 − 2 p1

Then there exists a constant C > 0 such that, for every continuous m-linear mapping A : Xp1 × · · · × Xpm → C, 1/ρ  +∞ X  ≤ CkAk. |A(ei1 , . . . , eim )|ρ  i1 ,...,im =1

Both approaches can be embedded into our framework, so that we are able to prove the following result. Theorem 1.2. Let m ≥ 1, let 1 ≤ s ≤ q ≤ 2, let p = (p1 , . . . , pm ) ∈ [1, +∞]m such that 1 1 1 − − ≥ 0. s q p Let also q1 , . . . , qm ∈ [λ, 2], where

λ=

1

1 2

+

1 s

−

Then the following are equivalent: (1) There is a constant Cq1 ...qm ≥ 1 such that 





P ∞  ∞  P   ...  i1 =1 i2 =1

1 q

. − p1

qm−2

∞ P

im−1 =1

∞ P

kA (ei1 , ..., eim ) kqqm

im =1

qm−1 ! qm−1 qm

for all continuous m-linear forms A : X p1 × · · · × Xpr → ℓs . 1 1 1 (2) q11 + · · · + q1m ≤ m 2 + s − q − p . When all the qi are equal, we get the following corollary:

1  q2  qq21 q1 q3     ≤ Cq1 ...qm kAk ···    


3

Corollary 1.3. Let m ≥ 1, let 1 ≤ s ≤ q ≤ 2, let p = (p1 , . . . , pm ) ∈ [1, +∞]m such that 1 1 1 ≥ 0. − − s q p

Let us define

ρ=

m+2

2m

1 s

−

1 q

. − p1

Then there exists a constant C > 0 such that, for every continuous m-linear mapping A : Xp1 × · · · × Xpm → ℓs , then 1/ρ  +∞ X  kA(ei1 , . . . , eim )kρq  ≤ CkAk. i1 ,...,im =1

It is plain that this corollary extends Defant and Sevilla-Peris result to the ℓp -case (and we get the same result if we choose p1 = · · · = pm = ∞). To show that Theorem 1.2 also implies Theorem 1.1 and the result of Praciano-Pereira, it suffices to choose q = 2, s = 1 and to consider only m-linear mappings which have their range in the span of the first basis vector. 2. A general interpolation method To prove our main theorem, we shall prove several particular cases and then deduce the result by interpolation. To avoid boring technicalities, let us introduce the following notation: let Y be a Banach space and let q = (q1 , . . . , qm ) ∈ [1, +∞)m . Then ℓq (Y ) denotes the Lorentz space ℓq1 (ℓq2 (. . . (ℓqm (Y )) . . . )). Proposition 2.1. Let Z, Y be Banach spaces, let N, m ≥ 1 and, for any k ∈ {1, . . . , N }, let q(k) ∈ [1, +∞)m . Suppose that, for any k ∈ {1, . . . , N }, T is a bounded Z into operator from

ℓq(k) (Y ) with norm less than C. Then, for any q ∈ (1, +∞)m such that q11 , . . . , q1m belongs to the convex hull of q11(k) , . . . , qm1(k) , k = 1, . . . , N , T is a bounded operator from Z to ℓq (Y ) with norm less than C. Proof. The proof is done by induction on N , using that, for any θ ∈ [0, 1], [ℓp (Y ), ℓq (Y )]θ = ℓr (Y )

with

1 r1

=

θ pi

+

1−θ qi

with θ1 + · · · + θN

(see [1]). The inductive step follows by associativity. Indeed, if we may write

1 θ1 θN = + ···+ qi qi (1) qi (N ) = 1, then we also have 1 − θN θN 1 = + qi pi qi (N )

setting

so that α1 + · · · + αN −1

α1 αN −1 θi 1 = + ···+ , αi = pi qi (1) qi (N − 1) 1 − θN = 1.

The previous proposition points out the necessity to describe certain convex hulls. Lemma 2.2. Let m ≥ 2, 0 < a < b and let, for any k ∈ {1, . . . , m}, Mk = (a, . . . , a, b, a . . . , a) where b is at the k-th position. Then the convex hull of M1 , . . . , Mm is the set of points (x1 , . . . , xm ) such that x1 + · · · + xm = (m − 1)a + b and xk ∈ [a, b] for any k ≥ 1. Proof. This follows from an easy induction.

4


As a consequence of these two results, we get the following corollary. Corollary 2.3. Let Z, Y be Banach spaces, let m ≥ 2 and let 1 ≤ λ ≤ 2. For any k ∈ {1, . . . , m}, let q(k) = (2, . . . , 2, λ, 2, . . . , 2) where λ is at the k-th position. Suppose that T maps continuously Z into ℓq(k) (Y ) with norm less than C. Then, for any q ∈ [λ, 2]m with 1 1 m−1 1 + ···+ = + , q1 qm 2 λ T maps continuously Z into ℓq (Y ). 3. Applications of the Minkowski’s inequality In this section, we shall prove the following proposition, which shows that an (ℓλ , ℓ2 )-mixed norm inequality implies many other inequalities of this kind. Proposition 3.1. Let Y be a Banach space, let m ≥ 1 and let λ ∈ [1, 2]. For any i ∈ {1, . . . , m}N , let xi ∈ Y . Define, for any k ∈ {1, . . . , m}, q(k) = (2, . . . , 2, λ, 2, . . . , 2). Assume that, for any σ a permutation of {1, . . . , m}, xσ(i) ∈ ℓq(1) (Y ). Then, for any k ∈ {1, . . . , m} and for any permutation σ of {1, . . . , m}, xσ(i) ∈ ℓq(k) (Y ) with kxσ(i) k ≤ max kxτ (i) kℓq(1) (Y ) . τ

Before to proceed with the proof, let us explain the notations. To say that xσ(i) belongs to ℓq(1) (Y ) simply means that 

1  22  λ2 λ  2  X  ∞ ∞ ∞ X X X     ∞      2      kxi k < +∞.   ···    ...    iσ(1) =1 iσ(2) =1  iσ(m−1) =1 iσ(m) =1 





 2  22

Since there are many 2/2 which simplify, we shall also denote the above quantity  λ/2 1/λ  +∞   X X < +∞, kxi k2    ik =1

ibk

ibk meaning that the sum is over all coordinates except the k-th coordinate.

Proof. We proceed by induction on k and assume that the property is true until rank k − 1. Since the assumptions are invariant by permutation, we may assume σ = Id. Then we just need to consider   12   λ2  λ2  ∞ ∞ ∞ ∞ X X X  X   2 kxi k   · · ·  · · · α=    . i1 =1

ik−1 =1

ik =1

ik+1 ,...,im =1

The integers i1 , . . . , ik−1 being fixed, we set



aik−1 ,ik = 

∞ X

ik+1 ,...,im =1

 λ2

2 kxi k  .


5

By Minkowski’s inequality applied in ℓ2/λ , +∞ X

ik−1 =1

Hence,

+∞ X

aik−1 ,ik

ik =1



!2/λ





 ≤



+∞ X

ik =1

∞ +∞ ∞ ∞ X X X  X · · · α≤   i1 =1

ik−2 =1

ik =1

 

+∞ X

ik−1 =1

∞ X

λ/2 2/λ  2/λ aik−1 ,ik   .

ik−1 =1 ik+1 ,...,im =1

By the induction hypothesis, applied to the transposition (k

 21  λ2  λ2   2 kxi k   · · ·   .

k − 1), α ≤ maxτ kxτ (i) kℓq(1) (Y ) .

4. A mixed (ℓλ , ℓ2 )-norm inequality We now give the main technical tool towards the proof of Theorem 1.2. It is a vector-valued version of the mixed (ℓ1 , ℓ2 )−norm inequality of Littlewood, allowing moreover ℓp -spaces instead of ℓ∞ . If Y is a Banach space of cotype 2, then C2 (Y ) denotes its cotype 2 constant. Proposition 4.1. Let X, Y be Banach spaces where Y has cotype 2. Let m, n, r ≥ 1, let p ∈ [1, +∞]m , let A : ℓnp1 × · · · × ℓnpm → X be m-linear and let v : X → Y be an (r, 1)-summing operator. Let us assume that   1 + ···+ 1 < 1 p1 pm r 1 1 1  For any l ≥ 1, P k6=l pk ≤ r − 2 .

Let us set

λ=

1 r

1 . − p1

Then, for any k ∈ {1, . . . , m},    λ2  λ1 √ m−1 X X  kvA(ei1 , . . . , eim k2   ≤ πr,1 (v)kAk. 2C2 (Y )  ik

ibk

√ Proof. For the sake of convenience, we shall denote M = ( 2C2 (Y ))m−1 πr,1 (v) and ai = kvA(ei1 , . . . , eim )k. We shall prove by induction on l ∈ [0, m − 1] the following statement Pl : For any p1 , . . . , pl ∈ [1, +∞]l with p11 +· · ·+ p1l ≤ r1 − 12 , for any A : ℓnp1 ×· · ·×ℓnpl ×ℓn∞ ×· · ·×ℓn∞ → X m-linear, setting 1 , λ= 1 1 1 − + · · · + r p1 pl

for any k ∈ {1, . . . , m}, (4.1)

λ/2 1/λ  X X 2   ≤ M kAk. ai   

ik

ibk

P0 is proved in [8, Lemma 2]. Let us assume that Pl−1 is true and let us prove Pl . For k ≤ l, let us fix x ∈ ℓnpk and let us consider Bk : ℓnp1 × · · · × ℓn∞ × · · · × ℓnpl × ℓn∞ × · · · × ℓn∞ → K

(z (1) , . . . , z (k) , . . . , z (m) ) 7→ A(z (1) , . . . , xz (k) , . . . , z (m) ).

6


By applying the induction hypothesis to Bk , we know that λ′ /2 1/λ′  X ′  X ≤ M kAkkxkℓnp a2i  |xik |λ    

k

ik

where we have set

ibk

l

1 1 X 1 = − . ′ λ r j=1 pj j6=k

We optimize this with respect to x describing the unit ball of ℓnpk . We get   λ′ ( pk′ )∗  λ1′ × ( pk1′ )∗ λ λ X X 2   ai ≤ M kAk   ik

where

pk ∗ λ′

ibk

pk λ′ .

denotes the conjugate exponent of 1

=1−

pk ∗ λ′

Now,

λ′ λ′ = . pk λ

This shows that (4.1) holds when k ≤ l. Let us also show that it also holds when k > l. The previous simple trick does not hold and the induction hypothesis applied to Bl and with l−1

1 X 1 1 = − ′ λ r j=1 pj now gives

(4.2)

∀x ∈ Bℓnp , l

X ik

Let us denote, for ik ∈ {1, . . . , n},

We intend to show that

P

ik

 

X ibk

λ′ /2

a2i |xil |2  

Sik = 

X ibk

′

1/2

a2i 

.

Siλk ≤ M λ kAkλ . We write X

Siλk =

ik

X

Siλ−2 k

ik

=

X

a2i

ibk

X X a2 i 2−λ S ik il b il

=

X X a2/s 2/s∗ i a 2−λ i S ik il b il

′

≤ M λ kAkλ .


7

where (s, s∗ ) is a couple of conjugate exponents. We then apply H¨ older’s inequality twice to get 1 1  s   s∗ X X a2 X X i    Siλk ≤ a2i  (2−λ)s S ik il ik ib ib l

l

1   st∗∗  t∗   1t   X X a2i   X X 2   ai ≤    . (2−λ)s Sik il il ib ib



t s

l

l

We then choose

s= so that

λ 2 − λ′ t∗ and ∗ = 2−λ s 2 t λ′ = . s λ

The inequality becomes X

Siλk

ik



 X X   ≤ il

ibl

λ

≤ (M kAk) t∗

  λ2  t1∗  λ′  1t  λ  X X 2   ai   

a2i  (2−λ′ ) Sik

il

ibl

 λ′  1t λ X X a2i    .  (2−λ′ ) Sik il ib 



l

It remains to control the last sum appearing on the right hand side of the inequality. By (the converse of) H¨ older’s inequality, it is sufficient to show that   X X a2 ′ i   |yil | ≤ (M kAk)λ (2−λ′ ) Sik il ibl ∗ for any y ∈ Bℓn λ ∗ . Since λλ′ = λpl′ , this means that we want to prove that ( λ′ ) X X a2 ′ ′ i |xil |λ ≤ (M kAk)λ 2−λ′ Sik il b il

for any x ∈ Bℓnp . Now, l

XX il

ibl

a2i

′

2−λ′

Sik

|xil |λ =

XX ik

by H¨ older’s inequality with

2−λ 2

+

′

λ 2

′

|xil |λ

ibk

ibk

= 1. To conclude, it remains to observe that P 2 X a2 ibk ai i = =1 Si2k Si2k ibk

and to use (4.2).

′ Si2−λ k

′  λ2′    2−λ 2 X X X a2 i    a2i |xil |2  ≤ 2 S ik i k

′

ibk

a2i

8


It remains to deduce the proposition from Pm−1 . The argument is exactly the same as we deduced Pl from Pl−1 , except that we do not need the second (and more difficult) part, since there are no k > m in {1, . . . , m}. This explains why we just need X 1 1 1 ∀l ∈ {1, . . . , m}, ≤ − pk r 2 k6=l

and not

m X 1 1 1 ≤ − . pk r 2

k=1

Remark 4.2. We cannot avoid the condition “for any l ≥ 1,

P

1 k6=l pk

≤

1 r

− 12 ”, see Section 6.4.

5. First part of the proof of the main theorem We now prove that (2) implies (1) in Theorem 1.2. Let X = ℓs , Y = ℓq and let W be the Banach space of continuous m-linear forms from Xp1 × · · · × Xpm to ℓs . By the Bennett-Carl inequalities ([5, 7]), the injection map v : ℓs → ℓq is (r, 1)-summing with 1r = 21 + 1s − 1q . Then λ=

1

1 2

+

1 s

−

1 q

= − p1

1 r

1 ≤ 2. − p1

Let finally T defined on W by T (A) = (A(ei1 , . . . , eim )) and q(k) = (2, . . . , 2, λ, 2, . . . , 2). Applying Propositions 4.1 and 3.1, T maps W into ℓq(k) for any k ∈ {1, . . . , m}. By our general interpolation procedure, T maps X into ℓq for any q = (q1 , . . . , qm ) ∈ [λ, 2]n with 1 m−1 1 m 1 1 1 1 + ··· + = + = + − − . q1 qm 2 λ 2 s q p Remark 5.1. If we take care of the constant in the Bohnenblust-Hille inequality, our method m−1 √ when K = C or Cm ≤ ( 2)m−1 when K = R. shows that it is valid with constant Cm ≤ √2π This constant comes from the best known constant in the mixed (ℓ1 , ℓ2 )-Littlewood inequality. In turn, this constant comes from the best constant in the Khintchine L1 − L2 -inequality. However, we know that the best constant in the Hille-Bohnenblust inequality is subpolynomial (see [16]). It would be nice to know if the constant in the mixed (ℓ1 , ℓ2 )-Littlewood inequality can also be chosen to be subpolynomial. 6. On the optimality 2m 6.1. A Kahane-Salem-Zygmund inequality. A way to prove that the exponent m+1 is optimal in the Bohnenblust-Hille inequality is to use the Kahane-Salem-Zygmund inequality, which allows to control the infinite norm of random polynomials. We need two variants of this inequality.

Lemma 6.1. Let d, n ≥ 1, p1 , . . . , pd ∈ [1, +∞]d and let, for p ≥ 1,   1 1 − if p ≥ 2 α(p) = 2 p  0 otherwise.

Then there exists a d-linear map A : ℓnp1 × . . . ℓnpd → C of form X (d) (1) ±zi1 · · · zid A(z (1) , . . . , z (d) ) = i1 ,...,id

such that

1

kAk ≤ Cd n 2 +α(p1 )+···+α(pd ) .


9

The proof follows from a straightforward modification of [4, Theorem 4] and is therefore omitted. We then also need a vector-valued version of this lemma. Lemma 6.2. Let d, n ≥ 1, s, p1 , . . . , pd ∈ [1, +∞]d+1 and let, for p ≥ 1,   1 1 − if p ≥ 2 α(p) = 2 p  0 otherwise.

Then there exists a d-linear map A : ℓnp1 × . . . ℓnpd → ℓnps of form X (d) (1) A(z (1) , . . . , z (d) ) = ±zi1 · · · zid eid+1 i1 ,...,id ,id+1

such that 1

∗

kAk ≤ Cd n 2 +α(p1 )+···+α(pd )+α(s ) . Proof. This lemma is an easy consequence of the previous lemma. Indeed, there is an isometric correspondence between d-linear maps ℓnp1 × · · · × ℓnpd → ℓns and (d + 1)-linear maps ℓnp1 × · · · × ℓnpd × ℓns∗ → C. The correspondence is given by     X X (d+1) (d) (1) (1) z 7→ ai1 ,...,id+1 zi1 · · · zid+1  . ai1 ,...,id+1 zi1 · · · zid eid+1  7→ z 7→ i1 ,...,id+1

i1 ,...,id+1

6.2. Optimality of Theorems 1.1 and 1.2. We just prove the optimality of Condition (2) in Theorem 1.2. Thus, let q1 , . . . , qm ∈ [λ, 2] satisfying (1) of Theorem 1.2. Let A : ℓnp1 ×· · ·×ℓnpm → ℓns be given by Lemma 6.2. Then 1

kAk ≤ Cm n 2 +

m+1 1 2 − p1

−···− p1m − s1∗

= Cm n 2 −| p |+ s m

1

1

whereas, for any i1 , . . . , im , kA(ei1 , . . . , eim )kq = n1/q . Then, n X

im =1

! qm−1 q m

kA(ei1 , . . . , eim )kqqm

=n

qm−1 qm

+

qm−1 q

.

It is then easy to show by induction that 





  ∞  ∞ ∞ P P  ...  P     jm−1 =1 j1 =1 j2 =1

∞ P

 ! qm−1 q m

kA (ej1 , ..., ejm ) kqqm

jm =1

qm−2 qm−1



For (1) of Theorem 1.2 to be true, it is necessary that 1 m 1 1 1 ≤ + ···+ − + . q qm 2 p s

q  1  qq2  q12 q1 3    1 1 1    = n q + q1 +···+ qm . ···     

10


6.3. The case m = 2. We now study more precisely the optimality in the Bohnenblust-Hille when m = 2 and K = R. Theorem 6.3. Let 1 ≤ p, q ≤ 2 with p1 + q1 ≤ 23 . Then for every continuous bilinear form A : c0 × c0 → R  p/q 1/p  ∞ ∞ X X 1 1   q  |A (ek , ej )|   ≤ 2 p + q −1 kAk  j=1

k=1

1

1

Moreover the constant 2 p + q −1 is optimal. Proof. If p1 + 1q = 1 < p, q < 2 and

3 2,

it has already been observed that the inequality holds true. Suppose now

3 1 1 + < . p q 2 Let 1 1 θ0 := 2 + −1 . p q Note that θ0 ∈ (0, 1) and let p0 , q0 be defined by 4p + 4q − 4pq p0 := , 2p + 4q − 3pq 4p + 4q − 4pq . q0 := 4p + 2q − 3pq We thus have 1 < p0 , q0 < 2 and 1 3 1 + = . p0 q0 2 So, from Part 1 of the proof we have  p0 /q0 1/p0  ∞ ∞ X X √    |A(ek , ej )|q0  ≤ 2kAk   k=1

j=1

for all continuous bilinear forms A : c0 × c0 → R. On the other hand it is well known that  1/2  ∞ ∞ X X   ≤ kAk. |A(ek , ej )|2  k=1

Since

j=1

(

1 p 1 q

= =

θ0 p0 θ0 q0

+ +

1−θ0 2 , 1−θ0 2 ,

from complex interpolation we have  p/q 1/p  ∞ ∞ √ θ0 X X    2 kAk ≤ |A(ek , ej )|q    k=1

j=1

1

1

= 2 p + q −1 kAk.

The case p, q ∈ {1, 2} with p1 + q1 < 23 can be verified straightforwardly. On the other hand, using ideas from [10], consider the bilinear form A(x, y) = x1 y1 + x1 y2 + x2 y1 − x2 y2


which satisfies kAk = 2. Then A verifies the equality  p/q 1/p  2 2 1 1 X X  |A (ek , ej )|q   ≤ 2 p + q −1 kAk  k=1

11

j=1

1

1

which shows that the constant 2 p + q −1 cannot be improved.

Remark 6.4. In the complex case, we obtain that for any continuous bilinear form A : c0 ×c0 → C,  p/q 1/p  1 + 1 −1 ∞ ∞ 2 p q X X  q kAk. |A (ek , ej )| ≤ √   π j=1 k=1

We have no information on the optimality of this estimate.

6.4. On the conditions of the mixed (ℓλ , ℓ2 )-norm inequality. We now show that we cannot avoid a condition like X 1 1 1 ∀l ∈ {1, . . . , m}, ≤ − pk r 2 k6=l

in the statement of Proposition 4.1. Indeed, let X = Y = C and v = Id, such that r = 1. Assume Pm−1 on the contrary that k=1 p1k > 12 . Let F : ℓnp1 × · · · × ℓnpm−1 → C be given by Lemma 6.1, so that m

kF k ≤ Cn 2

− p1 −···− p 1

1 m−1

. Let us define A : ℓnp1 × · · · × ℓnpm → C by (m)

A(z (1) , . . . , z (m) ) = z1

F (z (1) , . . . , z (m−1) ),

so that kAk = kF k. Then, for any θ > 0,  θ/2 1/θ  1/2  X X X    |A(ei1 , . . . , eim−1 , eim )|2   ≥ |A(ei1 , . . . , eim−1 , ei1 )|2  .  im

ic m

Now, we cannot have

ic m

1/2  X  |A(ei1 , . . . , eim−1 , ei1 )|2  ≤ CkAk ic m

since the left hand side is equal to n

m−1 2

and the right hand side behaves like n

m−1 1 1 2 + 2 − p1

−···− p

1 m−1

.

Acknowledgements. The authors would like to thank Prof. Fernando Cobos for fruitful conversations and his interest in this note. Part of this paper was written when the third named author was visiting Professors C. Barroso and E. Teixeira at the Department of Mathematics of Universidade Federal do Cear´ a, Brazil; he is very grateful for their hospitality. References [1] J. Bergh and J. L¨ ofstr¨ om, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. [2] O. Blasco, G. Botelho, D. Pellegrino, and P. Rueda, Summability of multilinear mappings: Littlewood, Orlicz and beyond, Monatsh. Math. 163 (2011), no. 2, 131–147. [3] H. P. Boas, The football player and the infinite series, Notices Amer. Math. Soc. 44 (1997), no. 11, 1430–1435. [4] H. P. Boas, Majorant series, J. Korean Math. Soc. 37 (2000), 321–337. [5] G. Bennett, Inclusion mappings between lp spaces, J. Functional Analysis 13 (1973), 20–27. [6] H. F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32 (1931), no. 3, 600–622.

12


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