Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant

ANNALES MATHÉMATIQUES

BLAISE PASCAL

Nacib Albuquerque, Gustavo Araújo, Mariana Maia, Tony Nogueira, Daniel Pellegrino & Joedson Santos

Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant

Volume 25, n^o1 (2018), p. 1-20.

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Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant

Nacib Albuquerque Gustavo Araújo

Mariana Maia Tony Nogueira Daniel Pellegrino

Joedson Santos

Abstract

Them-linear version of the Hardy–Littlewood inequality form-linear forms on`pspaces and m<p<2m,recently proved by Dimant and Sevilla-Peris, asserts that

©

­

­

­

«

∞ Õ

j_i=1 1≤i≤m

T ej₁, . . .,ejm

p p−m

ª

®

®

®

¬

p−m p

≤2^m−12 sup kx_ik≤1 1≤i≤m

|T(x1, . . .,xm) |

for all continuous m-linear formsT : `p× · · · ×`p → RorC. We prove a technical lemma, of independent interest, that pushes further some techniques that go back to the seminal ideas of Hardy and Littlewood. As a consequence, we show that the inequality above is still valid with 2^(m−1)/2replaced by 2(m−1)(p−m)/p. In particular, we conclude that form<p≤m+1 the optimal constants of the above inequality are uniformly bounded by 2; also, whenm=2,we improve the estimates of the original inequality of Hardy and Littlewood.

1. Introduction

Littlewood’s famous 4/3 inequality [19], proved in 1930, asserts that

©

­

«

∞

Õ

j,k=1

T(e_j,e_k)

4 3ª

®

¬

3 4

≤√

2 sup

kx1k,kx2k ≤1

|T(x₁,x₂)|,

for all continuous bilinear formsT:c₀×c₀ → C, and the exponent 4/3 cannot be improved. In some sense this result is the starting point of the theory of multiple summing operators (for recent results on summing operators we refer to [1, 8, 11, 15] and the references therein). From now on, for all Banach spacesE1, . . . ,Em,Fand allm-linear

N. Albuquerque is supported by CNPq, Grant 409938/2016-5, T. Nogueira is supported by Capes, D. Pellegrino and J. Santos are supported CNPq.

Keywords:Absolutely summing operators, Hardy–Littlewood inequalities, constants.

2010Mathematics Subject Classification:46G25, 47H60.

mapsT:E₁× · · · ×E_m→F,we denote kTk:= sup

kx1k,...,kxmk ≤1

kT(x₁, . . . ,xm)k.

Besides its own beauty, Littlewood’s insights motivated further important works of Bohnenblust and Hille (1931) and Hardy and Littlewood (1934). The Bohnenblust–Hille inequality [10] assures the existence of a constantB_m ≥1 such that

©

­

«

∞

Õ

j1,...,jm=1

T(e_j1, . . . ,e_jm)

2m m+1ª

®

¬

m+1 2m

≤B_mkTk,

for all continuous m–linear formsT: c₀× · · · ×c₀ → C. The case m = 2 recovers Littlewood’s 4/3 inequality. Three years later, using quite delicate estimates, Hardy and Littlewood [18] extended Littlewood’s 4/3 inequality to bilinear maps defined on

`<sub>p</sub>×`_q.In 1981, Praciano-Pereira [25] extended the Hardy–Littlewood inequalities to m-linear forms on`_pspaces forp≥2mand, quite recently, Dimant and Sevilla-Peris [17]

generalized the estimates for the case m < p < 2m. These results were extensively investigated in various directions in the recent years [2, 3, 5, 6, 7, 13, 14, 17, 21, 24]. As a matter of fact, all results hold for both real and complex scalars with eventually different constants.

From now on we denoteK=RorCand, for any function f, whenever it makes sense we formally define f(∞)=limp→∞ f(p); moreover, we adopt ^a₀ =∞for alla>0.

In general terms we have the followingm-linear inequalities:

• If 2m≤p≤ ∞, then there are constantsB^K_m,p ≥1 such that

©

­

« Õn

j1,...,jm=1

T(e_j1, . . . ,e_jm)

2m p m p+p−2mª

®

¬

m p+p−2m 2m p

≤B^K_m,pkTk

for allm-linear formsT :`<sub>p</sub><sup>n</sup>× · · · ×`ⁿ_p→Kand for all positive integersn.

• Ifm<p≤2m, then there are constantsB^K_m,p ≥1 such that

©

­

« Õn

j1,...,jm=1

T e_j1, . . . ,e_jm

p p−mª

®

¬

p−m p

≤B^K_m,pkTk

for allm-linear formsT :`<sub>p</sub><sup>n</sup>× · · · ×`ⁿ_p→Kand for all positive integersn.

The exponents of all above inequalities are optimal: if replaced by smaller exponents the constants will depend onn. However, looking at the above inequalities by an anisotropic viewpoint a much richer complexity arise (see, for instance, [2, 3, 5, 6, 13, 23]).

The investigation of the sharp constants in the above inequalities is more than a puzzling mathematical challenge; for applications in physics we refer to [20]. The first estimates forB^K_m,phad exponential growth; more precisely,

B^K_m,p ≤

√ 2m−1

,

for anym≥1. It was just quite recently that the estimates forB^K_m,pwere refined, see for instance [5, 6, 9] and references therein. It was proved in [9] that

B^R_m,∞< κ₁·m^{2−log 2−γ2} ≈κ₁·m^0.36482, B^C_m,∞< κ₂·m^1−γ2 ≈κ₂·m^0.21139,

for certain constants κ₁, κ₂ > 0, where γ is the Euler–Mascheroni constant, but, as conjectured in [24], these estimates seem to be suboptimal. For 2m(m−1)² <p<∞, among other results it was shown in [5] that we also have

B^R_m,p < κ₁·m^{2−log 2−γ2} ≈κ₁·m^0.36482, B^C_m,p < κ₂·m^1−γ2 ≈κ₂·m^0.21139.

The best known estimates ofB_m,^K_pfor the casem< p ≤2mare(√

2)^m−1 (see [3, 17]).

Recently, Cavalcante [12] has shown that these estimates are a straightforward consequence of a kind of regularity principle proved in [4], and also investigated monotonicity properties of the optimal constants. However, the search of the optimal constants in this setting (m<p≤2m) is still quite intriguing. For instance, ifp=mit is easy to show that the only Hardy–Littlewood type inequality

©

­

«

n

Õ

j1,...,jm=1

T(e_j1, . . . ,e_jm)

sª

®

¬

1 s

≤B^K_m,mkTk

happens for s=∞(of course, here we consider the sup norm in the left-hand-side of the inequality) and in this case it is obvious that the optimal constants areB_m,m^K =1. So, we have optimal constants equal to 1 forp=mand the best known constants equal to (√

2)^m−1for pclose tom. In this paper, among other results, we show that in fact the estimates(√

2)^m−1are far from being optimal: we prove that B^K_m,p ≤2

(m−1)(p−m)

p ,

and now, since 2^(m+

1)(p−m)

p −→1, we have a smooth connection between the estimates for p=mandp>m.From now on, forp=(p₁, . . . ,p_m) ∈ [1,∞]^mand 1≤k≤m, let

1 p _k := 1

p₁ +· · ·+ 1

p_k and 1 p :=

1 p _m

.

We present below the estimate obtained by Dimant and Sevilla-Peris [17] for further reference:

Theorem 1.1(Dimant and Sevilla-Peris). Let m ≥ 2be a positive integer andp = (p₁, . . . ,p_m) ∈ (1,∞]^mwith

1 2 ≤

1 p

<1.

Then

©

­

«

n

Õ

j1,...,jm=1

T e_j1, . . . ,e_jm

1 1−|¹p|ª

®

¬

1−

1 p

≤√

2m−1

kTk,

for allm-linear formsT :`<sub>p</sub><sup>n</sup><sub>1</sub>× · · · ×`ⁿ_pm→Kand all positive integersn. In particular, ifm<p≤2m, then

©

­

«

n

Õ

j1,...,jm=1

T e_j1, . . . ,e_jm

p p−mª

®

¬

p−m p

≤√

2 m−1

kTk

for allm-linear formsT :`<sup>n</sup><sub>p</sub>× · · · ×`_pⁿ→Kand all positive integersn.

The exponent 1−

1 p

⁻1

is optimal, but if one works in the anisotropic setting the result is not optimal (see, for instance, [7, 23]). The main results of the present paper are the forthcoming Theorems 3.3, 3.4 and 3.5 which also improve the original constants of the bilinear Hardy–Littlewood inequalities. As a consequence of these results, when m < p ≤ m+1, we shall prove that the optimal constants of the Hardy–Littlewood inequality are uniformly bounded by 2.

2. A multipurpose lemma

Let m ≥ 2 be a positive integer, F be a Banach space, A ⊂ I_m := {1, . . . ,m},p = (p₁, . . . ,p_m) ∈ [1,∞]^m,s, α≥1 and

B_p,s,α^A,F (n):=inf

















C(n) ≥0 :

©

­

­

«

n

Õ

ji=1

©

­

«

n

Õ

bji=1

T e_j1, . . . ,e_jm

sª

®

¬

1 sα

ª

®

®

¬

α1

≤C(n), for alli∈ A















 ,

in whichbj_imeans that the sum runs over all indexes but j_i, and the infimum is taken over all norm-onem-linear mapsT :`<sup>n</sup><sub>p1</sub>× · · · ×`_pⁿ_m→F. The following lemma, fundamental in the proof of our main results, is based on ideas dating back to Hardy and Littlewood (see [2, 16, 17, 18, 25]). It is our belief that it is a result of independent interest, with potential applications in the theory of multiple summing operators:

Lemma 2.1. Letp =(p₁, . . . ,p_m)andq =(q₁, . . . ,q_m)be such that1 ≤ p_k <q_k ≤

∞, k=1, . . . ,m, and also letλ₀,s≥1.

(1) If

s≥η₁ :=

1 λ0

− 1 p +

1 q

−1

>0, (2.1)

then

B_p,s,η^Im,F₁(n) ≤B_q,s,λ^Im,F

0(n).

(2) If

s≥η₂:=

1

λ₀ −

1 p m−1

+ 1 q m−1

−1

>0, then

B_p,s,η^{m},F₂ (n) ≤B_q,s,λ^Im,F

0(n).

Proof. To prove (1), lets, λ₀be such that (2.1) is fulfilled. Let us define λ_j :=

"

1

λ₀ −

1 p j

+ 1 q j

#−1

, j=1, . . . ,m.

Notice thatλ_m=η₁. Moreover, for allj =1, . . . ,m, we haveλ_j−1 < λ_j and q_jp_j

λ_j−1(q_j−p_j) ∗

= λ_j

λ_j−1, (2.2)

where the notation above denotes the conjugate number, i.e, ifa≥1,then 1/a+1/a^∗=1.

Let us suppose that, fork∈ {1, . . . ,m}, the inequality

©

­

­

«

n

Õ

ji=1

©

­

«

n

Õ

bji=1

T(e_j1, . . . ,e_jm)

sª

®

¬

1 sλk−1

ª

®

®

¬

λk−11

≤B_q,s,λ^Im,F

0(n)kTk

is true for all m–linear mapsT : `<sub>p</sub><sup>n</sup><sub>1</sub>× · · · ×`ⁿ_pk−1×`<sub>q</sub><sup>n</sup><sub>k</sub> × · · · ×`_qⁿ_m → F and for all i=1, . . . ,m. Let us prove that

©

­

­

«

n

Õ

ji=1

©

­

«

n

Õ

bji=1

T(e_j1, . . . ,e_jm)

sª

®

¬

1 sλk

ª

®

®

¬

λ1k

≤B_q,s,λ^Im,F

0(n)kTk (2.3)

for allm–linear mapsT :`<sup>n</sup><sub>p1</sub>× · · · ×`ⁿ_pk×`<sub>q</sub><sup>n</sup><sub>k+1</sub>× · · · ×`_qⁿ_m →Fand for alli=1, . . . ,m.

Consider

T :`<sup>n</sup><sub>p1</sub>× · · · ×`ⁿ_pk×`<sub>q</sub><sup>n</sup><sub>k+1</sub>× · · · ×`_qⁿ_m→F,

am-linear map and, for eachx∈B_`ⁿ

qk pk qk−pk

, defineT^(x):`<sup>n</sup><sub>p1</sub>×· · ·×`ⁿ_pk−1×`<sub>q</sub><sup>n</sup><sub>k</sub>×· · ·×`_qⁿ_m→F by

T^(x)

z⁽¹⁾, . . . ,z^(m)

=T

z⁽¹⁾, . . . ,z^(k−1),xz^(k),z^(k+1), . . . ,z^(m) ,

wherexz^(k)= xjz⁽_j^k)

n

j=1 ∈`ⁿ_pk. Observe that

kTk ≥sup

T^(x)

:x∈B_`ⁿ

qk pk qk−pk

.

By applying the induction hypothesis toT^(x), we get, for alli=1, . . . ,m,

©

­

­

« Õn

ji=1

©

­

« Õn

bji=1

T ej1, . . . ,ejm

s xjk

sª

®

¬

1 sλk−1

ª

®

®

¬

λk−11

=©

­

­

«

n

Õ

ji=1

©

­

«

n

Õ

bji=1

T e_j1, . . . ,e_jk−1,xe_jk,e_jk+1, . . . ,e_jm

sª

®

¬

1 sλk−1

ª

®

®

¬

λk−11

=©

­

­

«

n

Õ

ji=1

©

­

«

n

Õ

bji=1

T^(x) e_j1, . . . ,e_jm

s

ª

®

¬

1 sλk−1

ª

®

®

¬

λk−11

≤B_q,s,λ^Im,F

0(n) T^(x)

≤B_q,s,λ^Im,F

0(n)kTk.

By (2.2), using the characterization of the dual of`_p-type spaces, we have

©

­

­

«

n

Õ

jk=1

©

­

«

n

Õ

jbk=1

T e_j1, . . . ,e_jm

sª

®

¬

1 sλk

ª

®

®

¬

λ1k

=

©

­

­

«

©

­

«

n

Õ

jbk=1

T e_j1, . . . ,e_jm

sª

®

¬

1 sλk−1

ª

®

®

¬

n

jk=1

λk−11

h qk pk λk−1(qk−pk)

i∗

=©

­

­

­

«

sup

y∈B_`n qk pk λk−1(qk−pk)

n

Õ

jk=1

|y_jk|©

­

«

n

Õ

bjk=1

T e_j1, . . . ,e_jm

sª

®

¬

1 sλk−1

ª

®

®

®

¬

λk−11

.

We continue the proof by making some other changes on the arguments borrowed from [17, 18, 25] to encompass our relaxed hypotheses. The main difference is that we now work in a broader scenario withq_j ≤ ∞for allj, and for this task some technical modifications are in order. Since for all positive integersNand all scalarsw1, . . . ,w_N, we have

sup

y∈B_`N pν

ÕN

i=1

|w_i||y_i|= sup

x∈B_`N p

ÕN

i=1

|w_i||x_i|^ν, (2.4)

for all 1≤ν≤p<∞, we get

©

­

­

«

n

Õ

jk=1

©

­

«

n

Õ

jbk=1

T e_j1, . . . ,e_jm

sª

®

¬

1 sλk

ª

®

®

¬

λ1k

=©

­

­

­

«

sup

y∈B_`ⁿ qk pk λk−1(qk−pk)

Õn

jk=1

|y_jk|©

­

« Õn

jbk=1

T e_j1, . . . ,e_jm

sª

®

¬

1 sλk−1

ª

®

®

®

¬

λk−11

=©

­

­

­

« sup

x∈B_`n qk pk qk−pk

n

Õ

jk=1

|x_jk|^λk−1©

­

«

n

Õ

bjk=1

T ej1, . . . ,ejm

sª

®

¬

1 sλk−1

ª

®

®

®

¬

λk−11

= sup

x∈B_`ⁿ qk pk qk−pk

©

­

­

« Õn

jk=1

©

­

« Õn

bjk=1

T ej1, . . . ,ejm

s xjk

sª

®

¬

1 sλk−1

ª

®

®

¬

λk−11

≤B_q,s,λ^Im,F

0(n)kTk, (2.5)

and this concludes the proof of (2.3) fori =k. To prove (2.3) fori ,klet us initially considerk,mors> λ_m=η1. For eachi =1, . . . ,m, to simplify our notation, let us denote

Si :=©

­

« Õn

bji=1

kT(e_j1, . . . ,ejm)k^sª

®

¬

1 s

.

Note that Õn

ji=1

©

­

« Õn

bji=1

kT(e_j1, . . . ,e_jm)k^sª

®

¬

1 sλk

= Õn

ji=1

S_i^λk = Õn

ji=1

S_i^λk−sS_i^s= Õn

ji=1

Õn

bji=1

kT(e_j1, . . . ,e_jm)k^s S_i^s−λk

= Õn

jk=1

Õn

bjk=1

kT(e_j1, . . . ,e_jm)k^s S_i^s−λk

=

n

Õ

jk=1 n

Õ

bjk=1

kT(e_j1, . . . ,e_jm)k

s(s−λk) s−λk−1

S_i^s−λk

kT(e_j1, . . . ,e_jm)k

s(λk−λk−1) s−λk−1 .

From Hölder’s inequality with exponents r= s−λ_k−1

s−λ_k and r^∗= s−λ_k−1 λ_k −λ_k−1

we have

n

Õ

ji=1

©

­

«

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^sª

®

¬

1 sλk

=

n

Õ

jk=1 n

Õ

jbk=1

kT(e_j1, . . . ,e_jm)k

s(s−λk) s−λk−1

S_i^s−λk

kT(e_j1, . . . ,ejm)k

s(λk−λk−1) s−λk−1

≤

n

Õ

jk=1















©

­

«

n

Õ

jbk=1

kT(e_j1, . . . ,ejm)k^s S_i^s−λk−1

ª

®

¬

s−λk s−λk−1

©

­

«

n

Õ

jbk=1

kT(e_j1, . . . ,e_jm)k^sª

®

¬

λk−λk−1 s−λk−1 











 .

Now, by the Hölder inequality with exponents r= λ_k(s−λ_k−1)

λ_k−1(s−λ_k) and r^∗ =λ_k(s−λ_k−1)

s(λ_k−λ_k−1)

we have

n

Õ

ji=1

©

­

«

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^sª

®

¬

1 sλk

≤©

­

­

«

n

Õ

jk=1

©

­

«

n

Õ

bjk=1

kT(e_j1, . . . ,ejm)k^s S_i^s−λk−1

ª

®

¬

λk λk−1

ª

®

®

¬

λk−1 λk ·_s−λ^s−λk

k−1

ש

­

­

« Õn

jk=1

©

­

« Õn

jbk=1

kT(e_j1, . . . ,ejm)k^sª

®

¬

1 sλk

ª

®

®

¬

λ1k·^(λk_s−λ^−λk−1)s k−1

. (2.6)

Let us estimate separately the two factors of this product. It follows from the casei=k that

©

­

­

« Õn

jk=1

©

­

« Õn

jbk=1

kT(e_j1, . . . ,ejm)k^sª

®

¬

1 sλk

ª

®

®

¬

λ1k·^(λk_s−λ^−λk−1)s k−1

≤ B_q,s,λ^Im,F

0(n)kTk^{(λk−λk−1}

)s

s−λk−1 . (2.7)

In order to estimate the first factor of the product in (2.6), we first observe that, by (2.4), we obtain

©

­

­

«

n

Õ

jk=1

©

­

«

n

Õ

jbk=1

kT(e_j1, . . . ,ejm)k^s S_i^s−λk−1

ª

®

¬

λk λk−1

ª

®

®

¬

λk−1 λk

= sup

y∈B_`ⁿ qk pk λk−1(qk−pk)

Õn

jk=1

|y_jk| Õn

jbk=1

kT(e_j1, . . . ,e_jm)k^s S_i^s−λk−1

.

Straightforward computations give us

sup

y∈B_`n qk pk λk−1(qk−pk)

n

Õ

jk=1

|y_jk|

n

Õ

jbk=1

kT(e_j1, . . . ,e_jm)k^s S_i^s−λk−1

= sup

x∈B_`ⁿ qk pk qk−pk

Õn

jk=1

Õn

jbk=1

kT(e_j1, . . . ,e_jm)k^s S_i^s−λk−1

|x_jk|^λk−1

= sup

x∈B_`n qk pk qk−pk

n

Õ

ji=1 n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^s

S_i^s−λk−1 |x_jk|^λk−1

= sup

x∈B_`n qk pk qk−pk

n

Õ

ji=1 n

Õ

bji=1

kT(e_j1, . . . ,ejm)k^s−λk−1 S_i^s−λk−1

kT(e_j1, . . . ,e_jm)k^λk−1|x_jk|^λk−1.

By the Hölder inequality with exponents r= s

s−λ_k−1 and r^∗= s λ_k−1, we have

sup

x∈B_`n qk pk qk−pk

n

Õ

ji=1 n

Õ

bji=1

kT(e_j1, . . . ,ejm)k^s−λk−1 S_i^s−λk−1

kT(e_j1, . . . ,e_jm)k^λk−1|x_jk|^λk−1

≤ sup

x∈B_`n qk pk qk−pk

n

Õ

ji=1

©

­

«

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^s S_i^s

ª

®

¬

s−λk−1 s

ש

­

« Õn

bji=1

kT(e_j1, . . . ,e_jm)k^s|x_jk|^sª

®

¬

1 sλk−1

.

Since

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^s S_i^s =

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^s Ín

bji=1kT(e_j1, . . . ,e_jm)k^s =1,

we finally conclude that

©

­

­

«

n

Õ

jk=1

©

­

«

n

Õ

jbk=1

kT(e_j1, . . . ,e_jm)k^s S_i^s−λk−1

ª

®

¬

λk λk−1

ª

®

®

¬

λk−1 λk ·_s−λ^s−λk

k−1

≤ sup

x∈B_`n qk pk qk−pk

n

Õ

ji=1

©

­

«

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^s|x_jk|^sª

®

¬

1 sλk−1

= sup

x∈B_`ⁿ qk pk qk−pk

Õn

ji=1

©

­

« Õn

bji=1

T^(x) ej1, . . . ,ejm

s

ª

®

¬

1 sλk−1

≤ B_q,s,λ^Im,F

0(n)kTkλk−1

. (2.8)

Plugging (2.7) and (2.8) in (2.6), we obtain

n

Õ

ji=1

©

­

«

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^sª

®

¬

1 sλk

≤ B_q,s,λ^Im,F

0(n)kTkλk−1·_s−λ^s−λk k−1 ·

B_q,s,λ^Im,F

0(n)kTk^{(λk−λk−1)}

s s−λk−1

= B_q,s,λ^Im,F

0(n)λk−1·_s−λ^s−λk k−1 ·

B_q,s,λ^Im,F

0(n)^{(λk−λk−1)}

s s−λk−1

× kTk^λk−1·

s−λk s−λk−1 · kTk

(λk−λk−1)s s−λk−1

= B_q,s,λ^Im,F

0(n)λk

kTk^λk,

that is,

©

­

­

«

n

Õ

ji=1

©

­

«

n

Õ

bji=1

T(e_j1, . . . ,e_jm)

sª

®

¬

1 sλk

ª

®

®

¬

λ1k

≤B_q,s,λ^Im,F

0(n)kTk.

It remains to considerk=mands=λ_m=η₁. Fortunately, this case is simpler than the previous and we have

©

­

­

«

n

Õ

ji=1

©

­

«

n

Õ

bji=1

kT(e_j1, . . . ,e_jm)k^sª

®

¬

1 sη1

ª

®

®

¬

η11

=©

­

­

«

n

Õ

jm=1

©

­

«

n

Õ

cjm=1

kT(e_j1, . . . ,e_jm)k^sª

®

¬

1 sη1

ª

®

®

¬

η11

≤B_q,s,λ^Im,F

0(n)kTk, where the inequality is due to the casei=k.

The proof of (2) is similar, except for the last step (casek=m) where one may use an argument as (2.5), but it is somewhat predictable and we omit the proof.

Remark 2.2. The caseqk =∞for allk=1, . . . ,min (1) is known and follows the ideas from [18, 25]; our approach follows the lines of the modern presentation of [17].

3. Main results

We begin this section by recalling a particular instance of the Contraction Principle of [16, Theorem 12.2]. From now onri(t)are the Rademacher functions.

Lemma 3.1(Corollary of the Contraction Principle). LetNbe a positive integer, and AandBbe subsets of{1,. . . ,N}such thatA⊂B. Regardless of the choice of scalars a₁, . . . ,a_N,

∫ 1 0

Õ

i∈A

a_ir_i(t)

dt ≤

∫ 1 0

Õ

i∈B

a_ir_i(t)

dt.

The next lemma seems to be a by now standard consequence of the Contraction Principle (it was recently proved, for instance, in [22, Lemma 1]) but we present a proof here for the sake of completeness.

Lemma 3.2. Regardless of the choice of the positive integersm,Nand the scalarsa_i1,...,im, i₁, . . . ,i_m=1, . . . ,N, we have

ik=1,...,Nmax

k=1,...,m

a_i1,...,im ≤

∫

[0,1]^m

ÕN

i1,...,im=1

r_i1(t₁). . .r_im(t_m)a_i1,...,im

dt1. . .dtm. Proof. We will proceed by induction overm. For the casem =1 considerl ∈ B :=

{1, . . . ,N}andA:={l}. Thus, from Lemma 3.1,

∫ 1 0

Õ

i∈ {l}

a_ir_i(t)

dt≤

∫ 1 0

N

Õ

i=1

a_ir_i(t)

dt

and this implies

|a_l| ≤

∫ 1 0

N

Õ

i=1

a_ir_i(t)

dt for alll∈ {1, . . . ,N}. Hence,

i=1,...,maxN

|a_i| ≤

∫ 1 0

N

Õ

i=1

a_ir_i(t)

dt.

Let us suppose, as the induction step, that the result is valid form−1. For all positive integersi₁, . . . ,i_m,

∫

[0,1]^m

N

Õ

i1,...,im=1

ri1(t₁). . .rim(t_m)a_i1,...,im

dt1. . .dtm

=∫

[0,1]^m−1

"

∫ 1 0

ÕN

i1=1

ri1(t₁) × ÕN

i2,...,im=1

ri2(t₂). . .rim(t_m)a_i1,...,im

!

dt1

#

dt2. . .dtm

≥

∫

[0,1]^m−1

ÕN

i2,...,im=1

r_i2(t₂). . .r_im(t_m)a_i1,...,im

dt2. . .dtm

≥ a_i1,...,im

,

where we have used the casem=1 and the induction hypothesis on the first and second

inequalities, respectively. This concludes the proof.

Now we are able to prove our first main result, providing better constants for Theorem 1.1.

The main difference between the proof of our next result and the original proof of [17] is that here we use Lemma 3.2 in order to get better estimates.

Theorem 3.3. Let m ≥ 2be a positive integer and p = (p₁, . . . ,p_m) ∈ (1,∞]with

1 2 ≤

1 p

<1. Then, for allm-linear formsT : `<sub>p</sub><sup>n</sup><sub>1</sub> × · · · ×`ⁿ_pm → Kand all positive integersn,

©

­

«

n

Õ

j1,...,jm=1

T e_j1, . . . ,e_jm

1 1−|¹p|ª

®

¬

1−

1 p

≤2^(m−1)

1−

1 p

kTk.

In particular, ifm<p≤2m, then, for all continuousm-linear formsT :`<sub>p</sub>× · · · ×`_p→K

©

­

«

∞

Õ

j1,...,jm=1

T ej1, . . . ,ejm

p p−mª

®

¬

p−m p

≤2

(m−1)(p−m) p kTk.

Proof. LetS:`<sub>∞</sub><sup>n</sup>× · · · ×`_∞ⁿ →Kbe anm-linear form (note thatShas a different domain from the domain ofT). Considers=

1−

1 p

−1

. Sinces≥2, from Lemma 3.2, Hölder’s inequality and Khinchin’s inequality for multiple sums we have

n

Õ

j1=1

©

­

«

n

Õ

bj1=1

S e_j1, . . . ,e_jm

sª

®

¬

1 s

≤

n

Õ

j1=1

©

­

­

«

©

­

«

n

Õ

bj1=1

S e_j1, . . . ,e_jm

2ª

®

¬

1 2

ª

®

®

¬

2 s

max

bj1

S e_j1, . . . ,e_jm

!1−²_s

≤ Õn

j1=1

√ 2m−1

Rn

²_s R¹⁻

2 n s

! ,

where

Rn:=∫

I

Õn

bj1=1

rj2(t₂). . .rjm(t_m)S e_j1, . . . ,ejm

dt2. . .dtm

andI :=[0,1]^m−1. Since Õn

j1=1

√ 2m−1

R_n ²_s

R¹⁻

2 s n

!

=2^(m−1)

1−

1 p

×

n

Õ

j1=1

∫

I

n

Õ

bj1=1

rj2(t₂). . .rjm(t_m)S e_j1, . . . ,ejm

dt2. . .dtm

=2^(m−1)

1−

1 p

×

∫

I n

Õ

j1=1

S©

­

« e_j1,

n

Õ

j2=1

r_j2(t₂)e_j2, . . . ,

n

Õ

jm=1

r_jm(t_m)e_jmª

®

¬

dt2. . .dtm

≤2^(m−1)

1−

1 p

× sup

t2,...,tm∈[0,1]

n

Õ

j1=1

S©

­

« e_j1,

n

Õ

j2=1

r_j2(t₂)e_j2, . . . ,

n

Õ

jm=1

r_jm(t_m)e_jmª

®

¬

≤2^(m−1)

1−

1 p

× sup

t2,...,tm∈[0,1]

S©

­

«

·, Õn

j2=1

rj2(t₂)e_j2, . . . , Õn

jm=1

rjm(t_m)e_jmª

®

¬

≤2^(m−1)

1−

1 p

kSk,