E extracta mathematicae Vol. 33, Núm. 2, 145 – 147 (2018)

A Note on a Paper of S.G. Kim

Diana M. Serrano-Rodŕıguez

Departamento de Matemáticas, Universidad Nacional de Colombia

111321 - Bogotá, Colombia

dmserrano0@gmail.com , diserranor@unal.edu.co

Presented by Ricardo Garćıa Received August 20, 2018

Abstract: We answer a question posed by S.G. Kim in [3] and show that some of the results
of his paper are immediate consequences of known results.

Key words: Bohnenblust-Hille inequality.

AMS Subject Class. (2000): 46G25, 30B50.

The recent paper [3] deals with extreme multilinear forms and polynomials
and the constants of the Bohnenblust-Hille inequalities. In this note we answer
a question posed in [3] and show that two theorems stated in [3] are immediate
corollaries of well known results of this field.

Let K be R or C. The multilinear Bohnenblust-Hille inequality asserts
that, given a positive integer m, there is an optimal constant C (m : K) ≥ 1
such that 

 ∞∑
i1,...,im=1

∣∣U(ei1 , . . . , eim)∣∣ 2mm+1



m+1
2m

≤ C (m : K) ∥U∥ ,

for all bounded m-linear forms U : c0 × · · · × c0 → K. The case of com-
plex scalars was first investigated in [1] and the case of real scalars seems to
have been just explored more recently. It is well known that the exponent
2m/ (m + 1) is sharp, so one of the main goals of the research in this field is
to investigate the constants involved. The following result was proved in [4]:

Theorem 1. ([4, Corollary 5.4], 2018) Let m ≥ 2 be a positive inte-
ger. If the optimal constant C (m : R) is attained in a certain T : c0×· · ·×c0 →
R, then the quantity of non zero monomials of T is bigger than 4m−1 − 1.

As an immediate corollary we conclude that if N1, . . . , Nm ≥ 1 are positive
integers such that

m∏
j=1

Nj ≤ 4m−1 − 1 ,

145



146 d.m. serrano-rodŕıguez

then

sup


N1,...,Nm∑

i1,...,im=1

∣∣T (ei1 , . . . , eim)∣∣ 2mm+1



m+1
2m

< C (m : R) ,

where the sup runs over all norm one m-linear forms T : ℓN1∞ ×· · ·×ℓNm∞ → R.
In particular,

sup


 2∑

i,j,k=1

|T (ei, ej, ek)|
6
4




4
6

< C (3 : R) ,

where the sup runs over all norm one m-linear forms T : ℓ2∞ × ℓ2∞ × ℓ2∞ → R,
and this is the content of [3, Theorem 4.9].

The polynomial Bohnenblust-Hille inequality for real scalars asserts that,
given a positive integer m, there is an optimal constant Cp (m : R) ≥ 1 such
that ( ∑

|α|=m
|aα|

2m
m+1

)m+1
2m

≤ Cp (m : R) ∥Q∥ ,

for all N ≥ 1 and for all m-homogeneous polynomials Q : ℓN∞ (R) → R given
by

Q(z) =
∑

|α|=m
aαz

α.

To the best of our knowledge, the case of real scalars became unexplored
until the publication of the paper [2] in 2015, where it is proved that the
constants Cp (m : R) cannot be chosen with a sub-exponential growth. More
precisely,

Theorem 2. ([2, Theorem 2.2], 2015)

Cp (m : R) >

(
2

4
√
3

√
5

)m
> (1.177)

m
,

for all positive integers m ≥ 2.

The above result is, obviously, by far, rather precise than [3, Theorem 4.5],
which states that

Cp (m : R) ≥ 2
m+1
2m ,



a note on a paper of s.g. kim 147

for all positive integers m ≥ 2. The only case that deserves a little bit more
of attention is the case m = 2, since(

2
4
√
3

√
5

)2
< 2

3
4 ,

but in the case m = 2 a quick look at the proof of [2, Proof of Theorem 2.2]
shows that

Cp (2 : R) ≥
3

3
4

5
4

≈ 1. 823 6 > 2
3
4 ,

and this answers in the negative the Question (2) posed by the author in
[3, Question (2)].

References

[1] H.F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet
series, Ann. of Math. (2) 32 (1931), 600 – 622.

[2] J.R. Campos, P. Jiménez-Rodŕıguez, G.A. Muñoz-Fernández,
D. Pellegrino, J.B. Seoane-Sepúlveda, On the real polynomial
Bohnenblust-Hille inequality, Linear Algebra Appl. 465 (2015), 391 – 400.

[3] S.G. Kim, The geometry of L
(
3ℓ2∞

)
and optimal constants in the Bohnenblust-

Hille inequality for multilinear forms and polynomials, Extracta Math. 33 (1)
(2018), 51 – 66.

[4] D. Pellegrino, E. Teixeira, Towards sharp Bohnenblust-Hille constants,
Commun. Contemp. Math. 20 (2018), 1750029, 33 pp.