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

Linear Algebra and its Applications 526 (2017) 27–34

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Linear Algebra and its Applications www.elsevier.com/locate/laa

A new estimate for the constants of an inequality due to Hardy and Littlewood Antonio Gomes Nunes 1 Department of Mathematics - CCEN, UFERSA, Mossoró, RN, Brazil

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 13 January 2017 Accepted 25 March 2017 Available online 27 March 2017 Submitted by P. Semrl

In this paper we provide a family of inequalities, extending a recent result due to Albuquerque et al. © 2017 Elsevier Inc. All rights reserved.

MSC: 46G25 Keywords: Optimal constants Hardy–Littlewood inequality

1. Introduction The Hardy–Littlewood inequalities [11] for m-linear forms and polynomials (see [2–5, 7,9,13–15,17]) are perfect extensions of the Bohnenblust–Hille inequality [6] when the sequence space c0 is replaced by the sequence space p . These inequalities assert that for K K any integer m ≥ 2 there exist constants Cm,p , Dm,p ≥ 1 such that ⎛ ⎝

∞ 

⎞ mp+p−2m 2mp |T (ej1 , · · · , ejm )|

2mp mp+p−2m

⎠

j1 ,··· ,jm =1

1

E-mail addresses: [email protected], [email protected]. Partially supported by CNPq-Brazil, Grant 302834/2013-3.

http://dx.doi.org/10.1016/j.laa.2017.03.023 0024-3795/© 2017 Elsevier Inc. All rights reserved.

K ≤ Cm,p T  ,

(1.1)

A.G. Nunes / Linear Algebra and its Applications 526 (2017) 27–34

28

when 2m ≤ p ≤ ∞, and ⎛ ⎝

⎞ p−m p

∞ 

|T (ej1 , · · · , ejm )|

p p−m

⎠

K ≤ Dm,p T  ,

j1 ,··· ,jm =1

when m < p ≤ 2m, for all continuous m-linear forms T : p × · · · × p → K (here, and henceforth, K = R or C). Both exponents are optimal, i.e., cannot be smaller without paying the price of a dependence on n arising on the respective constants. Following usual convention in the field, c0 is understood as the substitute of ∞ when the exponent p goes to infinity. The investigation of the optimal constants of the Hardy–Littlewood inequalities is closely related to the fashionable, mysterious and puzzling investigation of the optimal Bohnenblust–Hille inequality constants (see, for instance [12,15] and the references therein). In this note we extend the following result of [1, Theorem 3]: Theorem 1 (Albuquerque et al.). Let m ≥ 2 be a positive integer and m < p ≤ 2m − 2. Then, for all continuous m-linear forms T : p × · · · × p → K and all positive integers n, we have ⎛ ⎜ ⎝

n 

ji =1

⎛ ⎝

n 

p ⎞ ⎞ p−(m−1) · p−m p p ⎟ |T (ej1 , . . . , ejm )| p−(m−1) ⎠ ⎠

p−m p

≤2

(m−1)(p−m+1) p

T  .

ji =1

More precisely, using a variant of the technique introduced in [1], we find a family of inequalities extending the above result. Our result reads as follows, where Aλ0 is the optimal constant of the Khinchin inequality (defined in Section 2): Theorem 2. If λ0 ∈ [1, 2) and λ0 m < p ≤

2λ0 (m − 1) , 2 − λ0

then ⎛ ⎜ ⎝

n 

ji =1

for

⎛ ⎝

n 

ji =1

⎞ 1s η1 ⎞ η11 −2(m−1) ⎟ s |T (ej1 , . . . , ejm )| ⎠ ⎠ ≤ Aλ0 s T 

A.G. Nunes / Linear Algebra and its Applications 526 (2017) 27–34

η1 = s=

29

λ0 p , p − λ0 m λ0 p , p − λ0 m + λ0

all m-linear forms T : np × · · · × np → K, and all positive integers n. Our main technique is based on an original argument developed in [1], with some slight technical changes. 2. The proof of Theorem 2 Let m ≥ 2 be a positive integer, F be a Banach space, A ⊂ Im := {1, . . . , m}, p1 , . . . , pm , s, α ≥ 1 and BpA,s,α,F,n 1 ,...,pm ⎧ ⎫ ⎛ ⎛ ⎞ 1s α ⎞ α1 ⎪ ⎪ ⎪ ⎪ n n ⎨ ⎬ ⎟ ⎜ ⎝ s⎠ := inf C(n) : ⎝ T (ej1 , . . . , ejm ) ⎠ ≤ C(n), for all i ∈ A , ⎪ ⎪ ⎪ ⎪ ji =1 ji =1 ⎩ ⎭ in which ji means that the sum runs over all indexes but ji , and the infimum is taken over all norm-one m-linear operators T : np1 × · · · × npm → F . We begin by recalling the following interesting technical lemma proved in [1]: Lemma 1. Let 1 ≤ pk < qk ≤ ∞, k = 1, . . . , m and λ0 , s ≥ 1. If  m   1 1 1 < − p q λ j j 0 j=1

⎡ and

s≥⎣

1 − λ0

m−1  j=1

⎤−1



1 ⎦ 1 − pj qj

=: η2

(2.1)

then ,s,λ0 ,F,n 1 ,F,n Bp{m},s,η ≤ BqI1m,...,q , 1 ,...,pm m

where ⎡

⎤  −1 m   1 1 ⎦ 1 η1 := ⎣ − − . λ0 j=1 pj qj We also need to recall the Khinchin inequality: for any 0 < q < ∞, there are positive constants Aq , Bq such that regardless of the positive integer n and of the scalar sequence (aj )nj=1 we have

A.G. Nunes / Linear Algebra and its Applications 526 (2017) 27–34

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 q ⎞ q1 ⎛ ⎞ 12  n  n     aj rj (t) dt⎠ ≤ Bq ⎝ |aj |2 ⎠ ,  [0,1]  j=1  j=1

⎛ ⎛ ⎞ 12  n  Aq ⎝ |aj |2 ⎠ ≤ ⎝ j=1

where rj are the classical Rademacher functions (random variables). The best constants Aq are the following ones (see [10]):

• Aq =

√

    q1 Γ 1+q √2 2 if q > q0 ∼ = 1.85; π

• Aq = 2 2 − q if q < q0 , where q0 ∈ (1, 2) is the only real number such that Γ √ π 2 . 1

1

 p0 +1  2

=

For complex scalars, using Steinhaus variables instead of Rademacher functions it is well known that a similar inequality holds, but with better constants. In this case the optimal constant is  • Aq = Γ

q+2 2

 q1

if q ∈ [1, 2].

The notation of the constants Aq above will be used in all this paper. The following result is a variant of a result proved in [1], and is based on the Contraction Principle (see [8, Theorem 12.2]). From now on ri (t) are the Rademacher functions. Lemma 2. Regardless of the choice of the positive integers m, N and the scalars ai1 ,...,im , i1 , . . . , im = 1, . . . , N , ⎛ max

ik =1,...,N k=1,...,m

⎜ |ai1 ,...,im | ≤ ⎝



[0,1]m

⎞1/t  t   N    ⎟  ri1 (t1 ) · · · rim (tm )ai1 ,...,im  dt1 · · · dtm ⎠ ,  i1 ,...,im =1 

for all t ≥ 1. 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 i1 , . . . , im , ⎛ ⎜ ⎝



[0,1]m

⎞1/t  t   N    ⎟  ri1 (t1 ) · · · rim (tm )ai1 ,...,im  dt1 · · · dtm ⎠  i1 ,...,im =1 

A.G. Nunes / Linear Algebra and its Applications 526 (2017) 27–34

⎛ ⎜

[0,1]m−1

⎛

⎞1/t  ⎛ ⎞t ⎞ 1t ⎤t  1   N N  ⎟  ⎢⎝  ⎥ ri1 (t1 ) ⎝ ri2 (t2 ) · · · rim (tm )ai1 ,...,im ⎠ dt1 ⎠ ⎦ dt2 · · · dtm ⎟ ⎣  ⎠ i1 =1  i2 ,...,im =1 ⎡⎛



= ⎜ ⎝

0

⎞1/t  t   N    ⎟  ri2 (t2 ) · · · rim (tm )ai1 ,...,im  dt2 · · · dtm ⎠  i2 ,...,im =1 



⎜ ≥⎝

31

[0,1]m−1

≥ |ai1 ,...,im | , where we used the Contraction Principle and the induction hypothesis on the first and second inequalities, respectively. This concludes the proof of the lemma. 2 Now we are able to complete the proof. Let S : n∞ × · · · × n∞ → K be an m-linear form and consider λ0 ∈ [1, 2) and ⎡ s=⎣

1 − λ0

m−1  j=1

⎤−1



1 ⎦ 1 − pj qj

≥ 2.

Since s ≥ 2, from Hölder’s inequality with θ = 2/s we obtain ⎛ ⎜ ⎝

n 

⎛

n 

⎝

j1 =1

⎞ 1s λ0 ⎞ λ10 ⎟ s |S (ej1 , . . . , ejm )| ⎠ ⎠

j 1 =1

⎛

⎛⎛

⎞λ0 ⎞ λ10 ⎞ 12 ⎞θ   1−θ n n ⎜⎜  ⎟ ⎟ ⎜ ⎟ 2 ⎜⎝⎝ ⎟ ⎟ . ≤⎜ |S (ej1 , . . . , ejm )| ⎠ ⎠ max |S (ej1 , . . . , ejm )| ⎝ ⎝ ⎠ ⎠ j ⎛

j 1 =1

j1 =1

1

Using Lemma 2 and Khinchin’s inequality for multiple sums ([16]) we have ⎛

⎛ ⎞ 1s λ0 ⎞ λ10 n n   ⎜ ⎟ s ⎝ |S (ej1 , . . . , ejm )| ⎠ ⎠ ⎝ j1 =1

⎛ ≤⎝

j 1 =1

n  

j1 =1

−(m−1) Rn Aλ0

! 2s

1− 2 Rn s

λ 0

⎞ λ1

0

⎠

A.G. Nunes / Linear Algebra and its Applications 526 (2017) 27–34

32

⎛ =A

−2(m−1) s λ0

⎞1/λ0  λ0  n    ⎟  rj2 (t2 ) · · · rjm (tm )S (ej1 , . . . , ejm ) dt2 · · · dtm ⎠  j=1 



n ⎜ ⎝

j1 =1 [0,1]m−1

⎛ −2(m−1) s

= Aλ 0



⎜ ⎝

1

⎞1/λ0  ⎛ ⎞λ0   n n     ⎟ S ⎝ej , rj2 (t2 )ej2 , . . . , rjm (tm )ejm⎠ dt2 · · · dtm⎠ 1    j1 =1 j2 =1 jm =1 n 

[0,1]m−1

⎛ ≤A

−2(m−1) s λ0

−2(m−1) s

≤ Aλ 0

 ⎛ ⎞λ0 ⎞1/λ0  n n n       ⎟ ⎜  ⎝ ⎠ sup rj2 (t2 )ej2 , . . . , rjm (tm )ejm  ⎠ ⎝ S ej1 , t2 ,...,tm ∈[0,1] j =1   j2 =1 jm =1 1 S ,

where ⎛



⎜ Rn := ⎝

[0,1]m−1

⎞ λ1  λ0 0  n    ⎟  rj2 (t2 ) · · · rjm (tm )S (ej1 , . . . , ejm ) dt2 · · · dtm ⎠ .  j=1  1

Repeating the same procedure for the other indexes we have ⎛

⎛ ⎞ 1s λ0 ⎞ λ10 n n   −2(m−1) ⎜ ⎟ s ⎝ |S (ej1 , . . . , ejm )| ⎠ ⎠ ≤ Aλ0 s S ⎝ ji =1

ji =1

for all i = 1, ..., m. Hence, from Lemma 1, we conclude that ⎛

⎛ ⎞ 1s η1 ⎞ η11 n n   −2(m−1) ⎜ ⎟ s ⎝ |T (ej1 , . . . , ejm )| ⎠ ⎠ ≤ Aλ0 s T  ⎝ ji =1

ji =1

for all m-linear forms T : np × · · · × np → K and all positive integers n, where ⎡

⎤  −1 m   1 1 1 ⎦ η1 := ⎣ − − . λ0 j=1 pj qj We thus conclude that if "

1 m−1 s= − λ0 p

#−1

and λ0 ∈ [1, 2)

≥2

A.G. Nunes / Linear Algebra and its Applications 526 (2017) 27–34

33

with p > λ0 m and η1 =

λ0 p , p − λ0 m

then ⎛ ⎜ ⎝

n 

ji =1

⎛ ⎝

n 

⎞ 1s η1 ⎞ η11 −2(m−1) ⎟ s |T (ej1 , . . . , ejm )| ⎠ ⎠ ≤ Aλ0 s T 

ji =1

for all m-linear forms T : np × · · · × np → K and all positive integers n. In other words, if λ0 ∈ [1, 2) with λ0 m < p ≤

2λ0 (m − 1) 2 − λ0

and λ0 p , p − λ0 m λ0 p s= , p − λ0 m + λ0

η1 =

then ⎛ ⎜ ⎝

n 

ji =1

⎛ ⎝

n 

⎞ 1s η1 ⎞ η11 −2(m−1) ⎟ s |T (ej1 , . . . , ejm )| ⎠ ⎠ ≤ Aλ0 s T 

ji =1

for all m-linear forms T : np × · · · × np → K and all positive integers n. Remark 1. If λ0 = 1 then we get ⎛ ⎜ ⎝

n 

ji =1

⎛ ⎝

n 

ji =1

⎞ 1s η1 ⎞ η11 −2(m−1) ⎟ s |T (ej1 , . . . , ejm )| ⎠ ⎠ ≤ A1 s T 

A.G. Nunes / Linear Algebra and its Applications 526 (2017) 27–34

34

with m < p ≤ 2m − 2 and p , p−m+1 p η1 = p−m s=

and we recover [1, Theorem 3]. Acknowledgement The author thanks the referees for important corrections and suggestions. References [1] N. Albuquerque, G. Araújo, M. Maia, T. Nogueira, D. Pellegrino, J. Santos, Optimal Hardy– Littlewood inequalities uniformly bounded by a universal constant, arXiv:1609.03081. [2] G. Araújo, D. Pellegrino, Lower bounds for the constants of the Hardy–Littlewood inequalities, Linear Algebra Appl. 463 (2014) 10–15. [3] G. Araújo, D. Pellegrino, D.D.P. Silva e Silva, On the upper bounds for the constants of the Hardy–Littlewood inequality, J. Funct. Anal. 267 (6) (2014) 1878–1888. [4] G. Araújo, D. Pellegrino, Lower bounds for the complex polynomial Hardy–Littlewood inequality, Linear Algebra Appl. 474 (2015) 184–191. [5] G. Araújo, D. Pellegrino, Optimal Hardy–Littlewood type inequalities for m-linear forms on p spaces with 1 ≤ p ≤ m, Arch. Math. (Basel) 105 (3) (2015) 285–295. [6] H.F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. 32 (1931) 600–622. [7] W. Cavalcante, D. Núñez-Alarcón, Remarks on an inequality of Hardy and Littlewood, Quaest. Math. 39 (2016) 1101–1113. [8] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995. [9] V. Dimant, P. Sevilla-Peris, Summation of coefficients of polynomials on p spaces, Publ. Mat. 60 (2) (2016) 289–310. [10] U. Haagerup, The best constants in the Khinchine inequality, Studia Math. 70 (1982) 231–283. [11] G. Hardy, J.E. Littlewood, Bilinear forms bounded in space [p, q], Quart. J. Math. 5 (1934) 241–254. [12] T. Nogueira, P. Rueda, Summability of multilinear forms on classical sequence spaces, Quaest. Math. (2017), in press. [13] D. Pellegrino, The optimal constants of the mixed (1 , 2 )-Littlewood inequality, J. Number Theory 160 (2016) 11–18. [14] D. Pellegrino, J. Santos, D. Serrano-Rodríguez, E.V. Teixeira, Regularity principle in sequence spaces and applications, arXiv:1608.03423 [math.CA]. [15] D. Pellegrino, E.V. Teixeira, Towards sharp Bohnenblust–Hille constants, Commun. Contemp. Math. (2017), http://dx.doi.org/10.1142/S0219199717500298, in press, arXiv:1604.07595v2. [16] D. Popa, Multiple Rademacher means and their applications, J. Math. Anal. Appl. 386 (2) (2012) 699–708. [17] T. Praciano-Pereira, On bounded multilinear forms on a class of p spaces, J. Math. Anal. Appl. 81 (2) (1981) 561–568.