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 References [1] N. Albuquerque, G. Arau´jo, M. Maia, T. Nogueira, D. Pellegrino, J. Santos, Optimal Hardy–Littlewood inequalitiesuniformlybounded byauniversalconstant, arXiv:1609.03081 [2] G.Arau´jo,D.Pellegrino,LowerboundsfortheconstantsoftheHardy-Littlewoodinequalities.LinearAlgebra Appl.463(2014), 10–15. [3] G. Arau´jo, D. Pellegrino, D. D. P. Silva e Silva, On the upper bounds for the constants of the Hardy– Littlewoodinequality.J.Funct. Anal.267(6)(2014), 1878-1888. [4] G. Arau´jo, D. Pellegrino, Lower bounds for the complex polynomial Hardy-Littlewood inequality. Linear AlgebraAppl.474(2015), 184–191. [5] G. Arau´jo, D. Pellegrino, Optimal Hardy-Littlewood type inequalities form-linear formson ℓp spaces with 1≤p≤m.Arch.Math.(Basel)105(2015), no.3,285–295. [6] H. F. 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Teixeira, Regularity principle in sequence spaces and applications,arXiv:1608.03423[math.CA]. [15] D.Pellegrino,E.V.Teixeira,Towards sharpBohnenblust–Hilleconstants, arXiv:1604.07595v2, toappear in Comm.Contemp.Math. [16] D.Popa,MultipleRademacher meansandtheirapplications.J.Math.Anal.Appl.386(2)(2012), 699-708. [17] T.Praciano-Pereira,Onboundedmultilinearformsonaclassofℓpspaces.J.Math.Anal.Appl.81(2)(1981), 561-568. (A.G.Nunes)Departmentof Mathematics-CCEN, UFERSA, Mossoro´, RN,Brazil. E-mail address: [email protected] and [email protected]