A new estimate for the constants of an inequality due to Hardy and Littlewood

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by  Antonio Gomes Nunes

A NEW ESTIMATE FOR THE CONSTANTS OF AN INEQUALITY DUE TO
HARDY AND LITTLEWOOD
ANTONIOGOMESNUNES
Abstract. Inthispaperweprovideafamilyofinequalities,extendingarecentresultdueto
Albuquerqueet al.
7
1
0
2 1. Introduction

b The Hardy–Littlewoodinequalities [12] for m–linear forms and polynomials(see [2, 3, 4, 5, 9,
e 15, 17]) are perfect extensions of the Bohnenblust–Hille inequality [6] when the sequence space
F
c is replaced by the sequence space ℓ . These inequalities assert that for any integer m 2
2 th0ere exist constants CK ,DK 1 supch that ≥
m,p m,p ≥
] mp+2mp−p2m
A ∞ 2mp K
F (1.1)  |T(ej1,··· ,ejm)|mp+p−2m ≤Cm,pkTk,
. j1,··X·,jm=1
h when 2m p , and 
t ≤ ≤∞
a p−m
m p
∞ p K
[  |T(ej1,··· ,ejm)|p−m ≤Dm,pkTk,

2 j1,··X·,jm=1
v whenm<p 2m,forallcontinuousm–linearformsT:ℓp ℓp K(here,andhenceforth,
0 K=R or C).≤Both exponents are optimal, i.e., cannot be s×m·a·l·l×er wi→thout paying the price of a
5 dependence on n arising on the respective constants. Following usual convention in the field, c
0
9
is understood as the substitute of ℓ when the exponent p goes to infinity.
1 ∞
The investigation of the optimal constants of the Hardy–Littlewood inequalities is closely
0
. related to the fashionable, mysterious and puzzling investigation of the optimal Bohnenblust–
1
Hille inequality constants (see, for instance [15] and the references therein).
0
In this note we extend the following result of [1, Theorem 3]:
7
1
Theorem 1 (Albuquerque et al.). Let m 2 be a positive integer and m<p 2m 2. Then,
v: for all continuous m-linear forms T :ℓ ≥ ℓ K, we have ≤ −
p p
i ×···× →
X p−m
p−(m−1) p p
r n n p p ·p−m (m−1)(p−m+1)
a   |T (ej1,...,ejm)|p−(m−1)  ≤2 p kTk.
jXi=1 jbXi=1
 
   
Moreprecisely,usingadifferenttechniquewefindafamilyofinequalitiesextendingtheabove
result. Our resultreadsasfollows,where A is the optimalconstantofthe Khinchininequality
λ0
(defined in Section 2):
Key words and phrases. Optimalconstants, Hardy–Littlewoodinequality.
PartiallysupportedbyCapes.
MSC2010: 46G25.
1
2 ANTONIOGOMESNUNES
Theorem 2. If λ [1,2] and
0
∈
2λ (m 1)
0
λ m<p − ,
0
≤ 2 λ
0
−
then
1
n n s1η1 η1
T (e ,...,e )s A−2(ms−1) T
  | j1 jm |   ≤ λ0 k k
jXi=1 jXbi=1
 
for    
λ p
0
η = ,
1
p λ m
0
−
λ p
0
s = ,
p λ m+λ
0 0
−
all m-linear forms T :ℓn ℓn K, and all positive integers n.
p ×···× p →
Our main technique is based on an original argument developed in [1], with some slight
technical changes.
2. The proof of Theorem 2
Letm 2beapositiveinteger,F beaBanachspace,A I := 1,...,m ,p ,...,p ,s,α
m 1 m
≥ ⊂ { } ≥
1 and
1
1α α
n n s
BA,s,α,F,n :=infC(n) : T (e ,...,e ) s C(n), for all i A,
p1,...,pm  jXi=1jbXi=1k j1 jm k   ≤ ∈ 
 
   
in which ji means that the sum runs over all indexes but ji, and the infimum is taken over all
norm-onem-linearoperatorsT :ℓn ℓn F. We beginbyrecallingthefollowinglemma
p1×···× pm →
proved inb[1]:
Lemma 1. Let 1 p <q , k =1,...,m and λ ,s 1. If
k k 0
≤ ≤∞ ≥
1
m m 1 −
1 1 1 1 − 1 1
(2.1) < and s =:η
2
p − q λ ≥λ − p − q 
j=1(cid:18) j j(cid:19) 0 0 j=1 (cid:18) j j(cid:19)
X X
then  
Bp{1m,.}..,,sp,mη1,F,n ≤BqI1m,.,.s.,,λqm0,F,n,
where
1
m −
1 1 1
η :=
1
λ − p − q 
0 j=1(cid:18) j j(cid:19)
X
 
WealsoneedtorecalltheKhinchininequality: forany0<q < ,therearepositiveconstants
∞
A , B such that regardless of the positive integer n and of the scalar sequence (a )n we have
q q j j=1
1 q 1 1
n 2 n q n 2
A a 2 a r (t) dt B a 2 ,
q | j|  ≤ (cid:12) j j (cid:12)  ≤ q | j| 
j=1 Z[0,1](cid:12)j=1 (cid:12) j=1
X (cid:12)X (cid:12) X
   (cid:12) (cid:12)   
where rj are the classical Rademacher fun(cid:12)ctions (ran(cid:12)dom variables).
(cid:12) (cid:12)
The best constants A are the following ones (see [11]):
q
HARDY–LITTLEWOOD CONSTANTS 3
1
Γ 1+q q
A =√2 2 if q >q =1.85;
• q (cid:0)√π (cid:1)! 0 ∼
1 1
Aq =22−q if q <q0,
•
whereq (1,2)istheonlyrealnumbersuchthatΓ p0+1 = √π. Forcomplexscalars,using
0 ∈ 2 2
Steinhaus variables instead of Rademacher functions it is well known that a similar inequality
(cid:0) (cid:1)
holds, but with better constants. In this case the optimal constant is
1
q+2 q
A =Γ if q [1,2].
q
• 2 ∈
(cid:18) (cid:19)
The notation of the constants A above will be used in all this paper. The following result is
q
a variantof [1], andis basedonthe ContractionPrinciple (see [8, Theorem12.2]). Fromnow on
r (t) are the Rademacher functions.
i
Lemma 2. Regardless of the choice of the positive integers m,N and the scalars a ,
i1,...,im
i ,...,i =1,...,N,
1 m
t 1/t
N
max a r (t ) r (t )a dt dt
ikk==11,,......,,mN| i1,...,im|≤Z[0,1]m(cid:12)(cid:12)(cid:12)i1,..X.,im=1 i1 1 ··· im m i1,...,im(cid:12)(cid:12)(cid:12) 1··· m
 (cid:12) (cid:12) 
for all t 1. (cid:12) (cid:12)
≥ (cid:12) (cid:12)
Proof. The proof is an adaptation of an argument used in [1]. Essentially, we have to use the
Contraction Principle inductively. The case m = 1 is nothing else than the standard version of
Contraction Principle (see [8, Theorem 12.2]). For all positive integers i ,...,i ,
1 m
t 1/t
N
r (t ) r (t )a dt dt
 (cid:12) i1 1 ··· im m i1,...,im(cid:12) 1··· m
Z[0,1]m(cid:12)(cid:12)i1,..X.,im=1 (cid:12)(cid:12)
 (cid:12)(cid:12)(cid:12) 1 N N (cid:12)(cid:12)(cid:12)  t 1t t 1/t
=Z[0,1]m−1Z0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)iX1=1ri1(t1)i2,..X.,im=1ri2(t2)t···rim(tm)ai11,/..t.,im(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt1  dt2···dtm
N
r (t ) r (t )a dt dt
≥Z[0,1]m−1(cid:12)(cid:12)(cid:12)i2,..X.,im=1 i2 2 ··· im m i1,...,im(cid:12)(cid:12)(cid:12) 2··· m
a , (cid:12) (cid:12) 
≥| i1,...,im| (cid:12)(cid:12) (cid:12)(cid:12)
where we used the Contraction Principle and the induction hypothesis on the first and second
inequalities, respectively. This concludes the proof of the lemma. (cid:3)
Now we are able to complete the proof. Let S :ℓn ℓn K be an m-linear form and
consider ∞×···× ∞ →
1
m 1 −
1 − 1 1
s= 2
λ − p − q  ≥
0 j=1 (cid:18) j j(cid:19)
X
 
and
λ [1,2].
0
∈
4 ANTONIOGOMESNUNES
Since s 2, from Lemma 2, Ho¨lder’s inequality and Khinchin’s inequality for multiple sums
≥
([16]), choosing θ =2/s we obtain
1
n n s1λ0 λ0
S(e ,...,e )s
  | j1 jm |  
jX1=1 jbX1=1
 
    1
1 θ λ0 λ0
n n 2 1 θ
≤  |S(ej1,...,ejm)|2  mbax|S(ej1,...,ejm)| −  
jX1=1 jbX1=1  (cid:18) j1 (cid:19)  
  1  
≤ n Aλ−0(m−1)Rn s2 Rn1−2s λ0λ0
jX1=1(cid:18)(cid:16) (cid:17) (cid:19)
  λ0 1/λ0
n n
−2(m−1)
=A s r (t ) r (t )S(e ,...,e ) dt dt
λ0 jX1=1Z[0,1]m−1(cid:12)(cid:12)(cid:12)(cid:12)jXb1=1 j2 2 ··· jm m j1 jm (cid:12)(cid:12)(cid:12)(cid:12) 2··· m
 n (cid:12)(cid:12) n n (cid:12)(cid:12) λ0  1/λ0
−2(m−1)
=A s S e , r (t )e ,..., r (t )e dt dt
λ0 Z[0,1]m−1jX1=1n(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)  j1 jX2=1nj2 2 j2 jXm=1njm m jm(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ02·1·/·λ0m
−2(m−1)
A s sup S e , r (t )e ,..., r (t )e
≤ λ0  (cid:12)  j1 j2 2 j2 jm m jm(cid:12) 
−2(m−1) t2,...,tm∈[0,1]jX1=1(cid:12)(cid:12)(cid:12)(cid:12)  jX2=1 jXm=1 (cid:12)(cid:12)(cid:12)(cid:12) 
A s S , (cid:12) (cid:12)
≤ λ0 k k
where
1
λ0 λ0
n
R := r (t ) r (t )S(e ,...,e ) dt dt .
n Z[0,1]m−1(cid:12)(cid:12)(cid:12)(cid:12)jbX1=1 j2 2 ··· jm m j1 jm (cid:12)(cid:12)(cid:12)(cid:12) 2··· m
Repeating the same procedu(cid:12)re for the other indexes we have (cid:12) 
(cid:12) (cid:12)
1
n n 1sλ0 λ0
S(e ,...,e )s A−2(ms−1) S
  | j1 jm |   ≤ λ0 k k
jXi=1 jbXi=1
 
   
for all i=1,...,m. Hence, from Lemma 1, we conclude that
1
n n 1sη1 η1
T (e ,...,e )s A−2(ms−1) T
  | j1 jm |   ≤ λ0 k k
jXi=1 jXbi=1
 
for all m-linear forms T :ℓn ℓn K and all positive integers n, where
p ×···× p →
1
m −
1 1 1
η := .
1
λ − p − q 
0 j=1(cid:18) j j(cid:19)
X
 
HARDY–LITTLEWOOD CONSTANTS 5
We thus conclude that if
1 m 1 −1
s= − 2
λ − p ≥
(cid:20) 0 (cid:21)
and
λ [1,2]
0
∈
with
p>λ m
0
and
λ p
0
η = ,
1
p λ m
0
−
then
1
n n 1sη1 η1
T(e ,...,e )s A−2(ms−1) T
  | j1 jm |   ≤ λ0 k k
jXi=1 jbXi=1
 
for all m-linear forms T :ℓn ℓn K and all positive integers n. In other words, if
p ×···× p →
λ [1,2]
0
∈
with
2λ (m 1)
0
λ m<p −
0
≤ 2 λ
0
−
and
λ p
0
η = ,
1
p λ m
0
−
λ p
0
s = ,
p λ m+λ
0 0
−
then
1
n n s1η1 η1
T(e ,...,e )s A−2(ms−1) T
  | j1 jm |   ≤ λ0 k k
jXi=1 jbXi=1
 
for all m-linear forms T :ℓn ℓn K and all positive integers n.
p ×···× p →
Remark 1. If λ =1 then we get
0
1
n n 1sη1 η1
T(e ,...,e )s A−2(ms−1) T
  | j1 jm |   ≤ 1 k k
jXi=1 jbXi=1
 
   
with m<p 2m 2 and
≤ −
p
s = ,
p m+1
−
p
η =
1
p m
−
and we recover [1, Theorem 3].
6 ANTONIOGOMESNUNES
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(A.G.Nunes)Departmentof Mathematics-CCEN, UFERSA, Mossoro´, RN,Brazil.
E-mail address: [email protected] and [email protected]