WHEN ARE THE HARDY-LITTLEWOOD INEQUALITIES CONTRACTIVE?
arXiv:1705.06307v1 [math.FA] 17 May 2017
W. CAVALCANTE, T. NOGUEIRA, D. PELLEGRINO, P. RUEDA, AND J. SANTOS
Abstract. The optimal constants of the m-linear Bohnenblust–Hille and Hardy–Littlewood inequalities are still
not known despite its importance in several fields of Mathematics. For the Bohnenblust–Hille inequality and real
scalars it is well-known that the optimal constants are not contractive. In this note, among other results, we show
that if we consider sums over M := M (m) indexes with M log M = o(m), the optimal constants are contractive. For
instance, we can consider
%
$
m
M =
1
1+
(log m) log log log m
where ⌊x⌋ := max{n ∈ N : n ≤ x}. In particular, if ε > 0 and M := M (m) ≤ m1−ε , then the Bohnenblust–Hille
inequality restricted to sums over M indexes is contractive.
1. Introduction
K
The Bohnenblust-Hille inequality [6] asserts that for all positive integers m ≥ 1 there is a constant Cm
≥ 1 such
that

 m+1
2m
∞
X
2m
m+1
K


(1)
T (ei1 , . . . , eim )
≤ Cm
kT k
i1 ,...,im =1
for all continuous m-linear forms T : c0 × · · · × c0 → K, where K = R or C. Above and henceforth, as usual, {ej }
are the canonical vectors of c0 and
kT k :=
sup
kx1 k,...,kxm k≤1
|T (x1 , . . . , xm )| ;
we also denote
eni := ei , n times
... , ei .
K
The optimal values of Cm
are unknown and good estimates for these constants are important for applications
K
(see [10]). The estimates for Cm
were strongly improved in recent years. The following table illustrates the original
and more recent estimates ([2, 4]):
1931
(2)
R
Cm
≤
C
Cm
≤
2014
√
m2
m−1
2
1.3m0.365
√
m2
m−1
2
m0.212 ,
The huge improvement in the upper bounds for the Bohnenblust–Hille constants was achieved by means of
“interpolations”, i.e., a clever usage of the Hölder inequality for mixed sums, technique that goes back to the works
of Luxemburg [9] and Benedeck–Panzone [5]. However, as commented in [11], these interpolative techniques seem
to be suboptimal, and in [11] it is even conjectured that the optimal constants of the Bohnenblust–Hille inequality
are universally bounded, i.e., universally bounded by a constant that does not depend on m.
For real scalars it is well known that
1
R
Cm
≥ 21− m ,
2010 Mathematics Subject Classification. 47H60, 47A63, 11Y60, 46G25.
Key words and phrases. Hardy–Littlewood inequalities; contractive constants; multilinear forms.
1
2
W. CAVALCANTE, T. NOGUEIRA, D. PELLEGRINO, P. RUEDA, AND J. SANTOS
so the Bohnenblust–Hille for real scalars is obviously non-contractive. As a consequence of the main results of this
paper we show that the Bohnenblust–Hille inequality is, however, somewhat “almost” contractive. More precisely,
we consider sums in less indexes, i.e.,
 m+1

2m
∞
X
2m
K
T en1 , . . . , enM m+1 

≤ Dm,M
kT k ,
i1
iM
i1 ,...,iM =1
and show that if the number of “blocks” M := M (m) is such that
lim
m→∞
M (m)
<∞
m1−ε
1
K
for some ε > 0, then limm→∞ Dm,M
= 1. As a matter of fact, we prove even more: if εm = log log
m , then
j
k
1
1− log log m
K
limm→∞ Dm,M = 1 for M (m) = m
, where ⌊x⌋ := max{n ∈ N : n ≤ x}. Note that if we could have
choosen ε = 0 then it would be contractive.
This note is organized as follows. In Section 2 we show when the Bohnenblust–Hille inequality is contractive. In
Section 3 we sharpen our techniques in order to deal with the Hardy Littlewood inequality. In both cases, we get
characterizations for the contractivity of the real constants.
2. When is the multilinear Bohnenblust–Hille inequality contractive?
It is well known that (for both real and complex scalars)
 21

∞
X

(3)
|T (ei1 , . . . , eim )|2  ≤ kT k
i1 ,...,im
for all continuous m-linear forms T : c0 × · · · × c0 → K. In fact, for every positive integer n, by the Khinchin
inequality for multiple sums [12, page 701] (since the constant of the Khinchin inequality in this case is 1) we have
1/2

2
 21

1
1
Z
Z
n
n
X
X



ri1 (t1 ) · · · rim (tm )T (ei1 , . . . , eim ) dt1 · · · dtm 
|T (ei1 , . . . , eim )|2  ≤  · · · i1 ,...,im
i1 ,...,im

=
0
0
Z1
Z1 · · · T
0
0
≤ kT k.
n
X
i1 =1
ri1 (t1 )ei1 , . . . ,
n
X
im =1
1/2
!2
rim (tm )eim dt1 · · · dtm 
The next theorem can be understood as a refinement of (1) and shows when inequalities of the type BohnenblustHille have contractive constants as the number of variables m increases. It is worth mentioning that if m increases,
the number of “blocks” M can be maintained constant or increased as a function of m. By M = M (m) we mean
that M can vary as a function of m, this trivially includes the case when M is kept constant.
Theorem 2.1. Let m, M be positive integers with M ≤ m and let n1 , . . . , nM ∈ {0, 1, . . . , m} with n1 +· · ·+nM = m.
Then

 m+1
2m
∞
X
2m
n1
nM m+1 
K M

(4)
T ei1 , . . . , eiM
≤ (CM
) m kT k
i1 ,...,iM =1
for all continuous m-linear forms T : c0 × · · · × c0 → K. Besides, if M = M (m) is so that limm→∞
K M
then limm→∞ (CM
) m = 1.
Proof. From [1] we know that
(5)


∞
X
i1 ,...,iM =1
+1
 M2M
2M
K
T en1 , . . . , enM M +1 
≤ CM
kT k
i1
iM
M log M
m
= 0,
WHEN ARE THE HARDY-LITTLEWOOD INEQUALITIES CONTRACTIVE?
3
for all continuous m–linear forms T : c0 × · · · × c0 → K. Since
1−θ
θ
1
2m = 2M +
2
m+1
M+1
with
M
,
m
by (a corollary of) the Hölder inequality, and using (3) and (5) we have

 m+1
2m
∞
X
2m
T (en1 , . . . , enM ) m+1 

i1
iM
θ=
i1 ,...,iM =1


≤ 
∞
X
i1 ,...,iM =1
M
+1  m 
 M2M
2M
 
T (en1 , . . . , enM ) M +1 
 
i1
iM
∞
X
i1 ,...,iM =1
M

+1  m
 M2M
X
2M


|T (eni11 , . . . , eniMM )| M +1 
≤ 
 kT k
 21 1− M
m
2

n
n
T (e 1 , . . . , e M )  
i1
iM
i1 ,...,iM
M
K m
≤ (CM
) kT k
K
and the inequality is proved. Besides, using the known estimates for CM
(see (2)) we have
M
M
K m
(CM
) ≤ αM β m
for suitable α, β > 0. Note that
lim
m→∞
αM β
if, and only if,
lim log
m→∞
if, and only if,
lim
m→∞
αM
M
m
β
=1
M
m
= 0,
M
(log (α) + β log M ) = 0.
m
This last equality is valid since
M log M
= 0,
m→∞
m
lim
and the proof is done.
Example 2.2. It is interesting to verify that
M=
$
m
1+ log log1 log m
(log m)
%
satisfies our hypotheses.
Degree
m
Number of blocks
m
M=
1+
(log m)
10100
∼ 1096
101000
∼ 10994
1010000
∼ 109993
10100000
∼ 1099992
1
log log log m
4
W. CAVALCANTE, T. NOGUEIRA, D. PELLEGRINO, P. RUEDA, AND J. SANTOS
Example 2.3. Another choice of M that fulfills the hypothesis is
k
j
1
M = m1− log log m .
This is interesting since it is written as M = m1−εm with limm→∞ εm = 0.
The next result gives a partial converse of the above theorem in the real case.
Proposition 2.4. Let ε ≥ 0 and let m, M be positive integers, M = M (m), such that M ≤ m1−ε . Let n1 , . . . , nM ∈
{0, 1, . . . , m} with n1 + · · · + nM = m. Let Km,M ≥ 1 be a constant such that

 m+1
2m
∞
X
2m
n
n
m+1
1
M


T ei1 , . . . , eiM
≤ Km,M kT k
i1 ,...,iM =1
for all continuous m-linear forms T : c0 × · · · × c0 → R. If limm→∞ Km,M = 1 then ε > 0 and therefore
M log M = o(m).
Proof. We first observe that by Theorem 2.1 such a constant Km,M exists. To prove the first assertion, let us
proceed by contradiction. Assume that ε = 0. Then M (m) has to be chosen less than or equal to m for each
m. Taking in particular M (m) = m, the inequality turns up to be the Bohnenblust-Hille inequality and then
R
Km,M ≥ Cm
. By [7] we have
1
R
Km,M ≥ Cm
≥ 21− m ,
for all m, which contradicts that limm→∞ Km,M = 1. Therefore, ε > 0. It remains to be proved that if ε > 0 then
M log M = o(m). Just consider
m1−ε log(m1−ε )
log m
M log(M )
.
≤
= (1 − ε)
m
m
mε
As the last term tends to 0 when m → ∞ the conclusion follows.
The main theorem of this Section summarizes the information we got so far on the contractivity of the constants
by combining both Theorem 2.1 and Proposition 2.4:
K
Theorem 2.5. Let Dm,M
≥ 1 be the smallest constant such that
 m+1

2m
∞
X
2m
K
T en1 , . . . , enM m+1 

≤ Dm,M
kT k
i1
iM
i1 ,...,iM =1
for all continuous m-linear forms T : c0 × · · · × c0 → K. Assume that M = M (m). Then,
K
K
is contractive in the sense that limm→∞ Dm,M
= 1. In particular, if
1. If M log M = o(m), then Dm,M
1−ε
K
M ≤m
for some fixed ε ≥ 0, then Dm,M is contractive.
2. If we choose m and M such that M ≤ m1−ε for some fixed ε ≥ 0 then the following assertions are equivalent
in the real case:
R
R
(a) Dm,M
is contractive, that is, limm→∞ Dm,M
= 1.
(b) ε > 0.
(c) M log M = o(m).
3. When is the multilinear Hardy–Littlewood inequality contractive?
The Hardy–Littlewood inequalities for m–linear forms (see [8, 13]) are in some sense natural extensions of the
Bohnenblust–Hille inequality when we replace c0 by ℓp . These inequalities assert that for any integer m ≥ 2 and
K
2m ≤ p ≤ ∞, there exists a constant Cm,p
≥ 1 such that,
 mp+p−2m

2mp
∞
X
2mp
K

(6)
|T (ej1 , . . . , ejm )| mp+p−2m 
≤ Cm,p
kT k ,
j1 ,...,jm =1
for all continuous m–linear forms T : ℓp × · · · × ℓp → K. The exponent
in (6) we recover the Bohnenblust–Hille inequality.
2mp
mp+p−2m
is optimal. Note that taking p = ∞
WHEN ARE THE HARDY-LITTLEWOOD INEQUALITIES CONTRACTIVE?
5
The constants of the Hardy–Littlewood inequality were investigated in recent papers (see [2] and the references
therein). In this section we investigate the inequality (6) allowing summability by blocks, in the lines of what was
done in the previous section with the Bohnenblust–Hille inequality. However, the appearance of the new parameter
p requires a refinement of the techniques previously used.
Theorem 3.1. For all positive integers m and M ≤ m, and all p > 2m, we have
 mp+p−2m

2mp
2
∞
X
2mp
M2(2m −4m+p) 2
n
n
K
mp+p−2m
1
M
2M m −2M m+mp−2m
T e , . . . , e


kT k
≤
C
M,p
i1
iM
i1 ,...,iM =1
for all continuous m-linear forms T : ℓp × · · · × ℓp → K.
Proof. The main result of [1] asserts that if 1 ≤ M ≤ m and n1 , . . . , nM ≥ 1 are positive integers such that
K
n1 + · · · + nM = m, then there is a constant DM,m
≥ 1 such that

(7)

∞
X
i1 ,...,iM =1
 M p+p−2m
2M p
2M
p
K
T en1 , . . . , enM M p+p−2m 
≤ DM,p
kT k
i1
iM
for all continuous m–linear forms T : ℓp × · · · × ℓp → K. Moreover, the exponent
also proved that
2Mp
Mp+p−2m
is optimal. In [1] it is
K
K
DM,p
≤ CM,p
(8)
K
for all 1 ≤ M ≤ m, where CM,p
is the optimal constant of the M -linear Hardy–Littlewood inequality.
It is obvious that
 p−2m+2

 p−2m+2

2p
2p
∞
∞
X
X
2p
2p
n
n
p−2m+2
1
M
p−2m+2
T e , . . . , e



≤
|T (ej1 , . . . , ejm )|
i1
iM
j1 ,...,jm =1
i1 ,...,iM =1
By [11, Lemma 5.1] we know that

(9)

for p > 2m. Thus, since
∞
X
j1 ,...,jm =1
1
2mp
mp+p−2m
=
|T (ej1 , . . . , ejm )|
2Mm2 −4Mm+Mp
2Mm2 −2Mm+mp−2m2
2Mp
Mp+p−2m
2p
p−2m+2
+
 p−2m+2
2p

1−
≤ kT k
2Mm2 −4Mm+Mp
2Mm2 −2Mm+mp−2m2
2p
p−2m+2
,
by (a corollary of) the Hölder inequality, and using (7), (8) and (9) we have
 mp+p−2m

2mp
∞
X
2mp
2M2m2 −4M m+M p 2
n
n
K
mp+p−2m
1
M
2M m −2M m+mp−2m
T e , . . . , e


kT k .
≤
C
M,p
i1
iM
i1 ,...,iM =1
The next theorem shows when the constants of the previous theorem are contractive.
Theorem 3.2. If for each positive integer m we consider M = M (m) ≥ 2 and p = p(m) such that
lim
m→∞
M (m) log M (m)
=0
m
and
for all m, then
2m(m − 1)2 < p(m)
(
M 2m2 −4m+p
K
lim CM,p
m→∞
)
2M m2 −2M m+mp−2m2
= 1.
6
W. CAVALCANTE, T. NOGUEIRA, D. PELLEGRINO, P. RUEDA, AND J. SANTOS
Proof. Since 2m(m − 1)2 < p(m) for all m, by [2, Theorem 3.1] there exists a constant α > 0 such that
K
≤ α(M (m))0.365
CM(m),p(m)
for all m.
Let us see that
(
M 2m2 −4m+p
lim
m→∞
αM 0.365
)
2M m2 −2M m+mp−2m2
=1
where M = M (m) and p = p(m). Indeed, if β > 0 then
(
M 2m2 −4m+p
lim
m→∞
β
αM
β
αM
if, and only if,
lim log
m→∞
if, and only if,
)
2M m2 −2M m+mp−2m2
=1
2m2 −4m+p)
2M mM2(−2M
m+mp−2m2
!
= 0,
M 2m2 − 4m + p
lim
(log (α) + β log M ) = 0
m→∞ 2M m2 − 2M m + mp − 2m2
Dividing and multiplying the above quotient by mp we get:
2
+
1
2 mp − 4 m
p
M log (α) + βM log M
lim
=0
2
m→∞ 2 M m − 2 M m + 1 − 2 m
m
m p
m p
p
and using that
conclusion.
m
p
≤
m2
p
=
m2
p(m)
→ 0 (recall that p(m) > 2m(m − 1)2 ) and limm→∞
M log M
m
= 0 we get the
Although the proof of [2, Theorem 3.1] strongly requires the condition p > 2m(m − 1)2 , we shall see in the next
m2
= 0 under
result that Theorem 3.2 works when this condition is weakened by the new condition limm→∞ p(M(m))
the mild assumption of (M (m))m and (p(m))m being increasing. Actually, the new condition is the weakest one as
it leads up to a characterization of the contractivity of the constants in the real case.
Theorem 3.3. If M = M (m) ≥ 2 and p = p(m) are such that the sequence (p(m))m is non decreasing,
M(m)
m2
limm→∞ M(m) log
= 0 and limm→∞ p(M(m))
= 0, then
m
(
M 2m2 −4m+p
K
lim CM,p
m→∞
)
2M m2 −2M m+mp−2m2
=1
where M = M (m), p = p(mM(m) ) and the integer mM(m) is chosen such that 2M (m)(mM(m) )2 ≤ p(mM(m) ).
2
2
2
m
m
m
Proof. Since the sequence (p(m))m is non decreasing, p(m)
≤ p(M(m))
. Then limm→∞ p(m)
= 0. This last limit
∞
2
guarantees the existence of a sequence (mk )k=1 such that mk ≥ k and 2kmk ≤ p(mk ) for all k. Hence, for each k,
2k(k − 1)2 < 2km2k ≤ p(mk ).
Thus, we can apply [2, Theorem 3.1] and therefore there exists a constant α > 0 such that
K
≤ αk 0.365
Ck,p(m
k)
for every k ≥ 2. Since this is true for every k ≥ 2, letting k = M (m) this last inequality reads as
K
≤ α(M (m))0.365
CM(m),p(m
M (m) )
for all m.
Similarly to the proof of Theorem 3.2 we see that
(
M 2m2 −4m+p
lim
m→∞
αM 0.365
)
2M m2 −2M m+mp−2m2
=1
where M = M (m) and p = p(mM(m) ). We include details for the sake of completeness. If β > 0 then
(
M 2m2 −4m+p
lim
m→∞
αM
β
)
2M m2 −2M m+mp−2m2
=1
WHEN ARE THE HARDY-LITTLEWOOD INEQUALITIES CONTRACTIVE?
7
if and only if
(
M 2m2 −4m+p
lim log
β
αM
m→∞
if and only if
)
2M m2 −2M m+mp−2m2
!
= 0,
M 2m2 − 4m + p
(log (α) + β log M ) = 0.
lim
m→∞ 2M m2 − 2M m + mp − 2m2
Dividing and multiplying the above quotient by mp we get:
2
+
1
2 mp − 4 m
p
M log (α) + βM log M
=0
lim
2
m→∞ 2 M m − 2 M m + 1 − 2 m
m
m p
m p
p
and using that
m
p
≤
m2
p
=
m2
p(mM (m) )
≤
m2
p(M(m))
→ 0 and limm→∞
M log M
m
= 0 we get the conclusion.
Remark 3.4. If we assume that (M (m))m and (p(m))m are increasing sequences (which is not really restrictive in
our study) then Theorem 3.2 is a consequence of Theorem 3.3 as we are going to prove.
Proof. Let us consider the sequence (q(k))k given by q(k) := p(m) for every k ∈ [M (m−1), M (m)−1], m = 1, 2, 3, . . .
(here M (0) = 1). Note that q(k) is well defined because (M (m))m is increasing. Since
(m − 1)2
1
(m − 1)2
(m − 1)2
(m − 1)2
=
=
=
<
2
q(M (m − 1))
q(M (m) − 1)
p(m)
2m(m − 1)
2m
we have that limm→∞
m2
q(M(m))
= 0. By Theorem 3.3 it follows that
(
M 2m2 −4m+q
K
lim CM,q
m→∞
(10)
)
2M m2 −2M m+mq−2m2
=1
where M = M (m), q = q(mM(m) ) and the integer mM(m) is chosen such that 2M (m)(mM(m) )2 ≤ q(mM(m) ).
Actually, we can choose mM(m) equal to M (m) − 1 because
2M (m)(M (m) − 1)2 < p(M (m)) ≤ p(m) = q(M (m) − 1).
Therefore, the limit in (10) can be rewritten as
(
M 2m2 −4m+p
K
lim CM,p
m→∞
where M = M (m) and p = p(m).
)
2M m2 −2M m+mp−2m2
=1
Now we have a converse of Theorem 3.3:
Proposition 3.5. Let ε ≥ 0 and let m ≥ 2, M be positive integers. Assume that p = p(m) > 2m and M = M (m)
are such that M (m) ≤ m1−ε and m2 ≤ p(M (m))1−ε . Let n1 , . . . , nM ∈ {0, 1, . . . , m} with n1 + · · · + nM = m. Let
Km ≥ 1 be a constant such that
 mp+p−2m

2mp
∞
X
2mp
n
n
mp+p−2m
1
M
T e , . . . , e


≤ Km kT k
i
i
1
M
i1 ,...,iM =1
for all continuous m-linear forms T : ℓp × · · · × ℓp → R.
M(m)
m2
limm→∞ M(m) log
= 0 and limm→∞ p(M(m))
= 0.
m
If limm→∞ Km = 1 then ε > 0 and therefore
Note that in Proposition 3.5 M (m) cannot be constant as m → ∞.
Proof. The proof follows the lines of the proof of Proposition 2.4. We include it for the sake of completeness. By
Theorem 3.1 such a constant Km exists. Assume by contradiction that ε = 0. Then M (m) ≤ m and p(M (m)) ≥ m2
for each m. When M (m) = m the inequality becomes the classical Hardy-Littlewood inequality and then Km ≥
R
Cm,p
. By [3, Theorem 1.2] we have
2
for all m. Taking now p(m) = m we get
R
Km ≥ Cm,p
≥2
Km ≥ 2
(m−1)(p−2m)
mp
(m−1)(m2 −2m)
m3
=2
,
(m−1)(m−2)
m2
8
W. CAVALCANTE, T. NOGUEIRA, D. PELLEGRINO, P. RUEDA, AND J. SANTOS
which contradicts that limm→∞ Km = 1. Therefore, ε > 0. In this case,
p(M (m))1−ε
1
m2
.
≤
=
p(M (m))
p(M (m))
p(M (m))ε
As the last term tends to 0 when m → ∞ the conclusion follows.
The following theorem summarizes the information we got so far combining Theorems 3.1 and 3.3, and Proposition
K
3.5. Let m, M be positive integers, p > 2m and n1 , . . . , nM ∈ {0, 1, . . . , m} with n1 + · · · + nM = m. Let Dm,M,p
≥1
be the smallest constant such that


∞
X
i1 ,...,iM =1
 mp+p−2m
2mp
2mp
K
T en1 , . . . , enM mp+p−2m 
≤ Dm,M,p
kT k
i1
iM
for all continuous m-linear forms T : ℓp × · · · × ℓp → K. By Theorem 3.1 we know that
(
M 2m2 −4m+p
K
Dm,M,p
≤
K
CM,p
)
2M m2 −2M m+mp−2m2
.
Theorem 3.6. Assume that M = M (m) ≥ 2 and p = p(m) > 2m. Then,
K
K
=
1. If M log M = o(m) and p(m) > 2m(m−1)2 , then Dm,M,p
is contractive in the sense that limm→∞ Dm,M,p
1.
m2
2. If M log M = o(m) and the sequence (p(m))m is non decreasing such that limm→∞ p(M(m))
= 0, then
K
Dm,M,p is contractive.
3. If we choose m and M be such that M ≤ m1−ε and m2 ≤ p(M (m))1−ε for some fixed ε ≥ 0 then the
following assertions are equivalent in the real case:
R
R
is contractive, that is, limm→∞ Dm,M,p
= 1.
(a) Dm,M,p
(b) ε > 0.
m2
= 0.
(c) M log M = o(m) and limm→∞ p(M(m))
Acknowledgment: P. Rueda and D. Pellegrino are supported by Ministerio de Economı́a y Competitividad
and FEDER under project MTM2016-77054-C2-1-P. This work was done while P. Rueda was visiting the Department of Mathematical Sciences at Kent State University suported by Ministerio de Educación, Cultura y Deporte
PRX16/00037. She thanks this Department for its kind hospitality.
References
[1] N. Albuquerque, G. Araújo, D. Núñez-Alarcón, D. Pellegrino, and P. Rueda, Bohnenblust–Hille and Hardy–Littlewood inequalities
by blocks, arXiv:1409.6769v6 [math.FA].
[2] G. Araujo, D. Pellegrino, On the constants of the Bohnenblust–Hille and Hardy–Littlewood inequalities, Bull. Braz. Math. Soc.
(N.S.) 48 (2017), no. 1, 141–169.
[3] G. Araujo, D. Pellegrino, Lower bounds for the constants of the Hardy-Littlewood inequalities. Linear Algebra Appl. 463 (2014),
10–15.
p
[4] F. Bayart, D. Pellegrino and J. B. Seoane-Sepúlveda, The Bohr radius of the n–dimensional polydisk is equivalent to (log n)/n,
Adv. Math. 264 (2014), 726–746.
[5] A. Benedek, R. Panzone, The space Lp , with mixed norm, Duke Math. J. 28 1961 301–324.
[6] H. F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. 32 (1931), 600–622.
[7] D. Diniz, G. Munoz-Fernandez, D. Pellegrino, J. Seoane-Sepulveda, Lower bounds for the constants in the Bohnenblust-Hille
inequality: the case of real scalars. Proc. Amer. Math. Soc. 132 (2014), 575–580.
[8] G. Hardy and J. E. Littlewood, Bilinear forms bounded in space [p, q], Quart. J. Math. 5 (1934), 241–254.
[9] W.A.J. Luxemburg, Banach Function Spaces, Essen, 1955.
[10] A. Montanaro, Some applications of hypercontractive inequalities in quantum information theory, J. Math. Physics 53 (2012).
[11] D. Pellegrino, E. Teixeira, Towards sharp Bohnenblust–HIlle constants, to appear in Comm. Contemp. Math.
[12] D. Popa, Multiple Rademacher means and their applications. J. Math. Anal. Appl. 386 (2012), no. 2, 699–708.
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WHEN ARE THE HARDY-LITTLEWOOD INEQUALITIES CONTRACTIVE?
9
(W. Cavalcante) Departamento de Matemática - Federal University of Pernambuco - Recife - Brazil
E-mail address: [email protected]
(T. Nogueira) Departamento de Matemática, Universidade Federal da Paraı́ba, 58.051-900 - João Pessoa, Brazil, and Departamento de Ciências Exatas Tecnológicas e Humanas, Universidade Federal Rural do Semi-Árido, 59.515-000 - Angicos,
Brazil.
E-mail address: [email protected] and [email protected]
(D. Pellegrino) Departamento de Matemática, Universidade Federal da Paraı́ba, 58.051-900 - João Pessoa, Brazil.
E-mail address: [email protected] and [email protected]
(P. Rueda) Departamento de Análisis Matemático,
E-mail address: [email protected]
(J. Santos) Departamento de Matemática,
E-mail address: [email protected]
Universidad de Valencia,
Universidade Federal da Paraı́ba,
46100 Burjassot, Valencia.
58.051-900 - João Pessoa, Brazil.