\section{Introduction}

If $E$ is a Banach space, real or complex, we say that $P$ is a homogeneous
polynomial on $E$ of degree $m\in{\mathbb{N}}$ if there exists a symmetric
$m$-linear form on $E^{m}$ such that $P(x)=L(x,\ldots,x)$ for all $x\in E$. We
denote by ${\mathcal{P}}(^{m}E)$ and ${\mathcal{L}}^{s}(^{m}E)$ the spaces of
continuous $m$-homogeneous and continuous symmetric $m$-linear forms on $E$.
It is well-known that a homogeneous polynomial is continuous if and only if
$P$ is bounded over the unit ball ${\mathsf{B}}_{E}$ of $E$, and in that case
\[
\|P\|=\sup\{|P(x)|:x\in{\mathsf{B}}_{E}\}
\]
defines a norm in ${\mathcal{P}}(^{m}E)$. If $P\in{\mathcal{P}}(^{m}E)$, we
shall refer to $\|P\|$ as the polynomial norm of $P$ in $E$. This norm is very
difficult to compute in most cases, for which reason it would be interesting
to obtain reasonably good estimates on it. The $\ell_{p}$ norm of the
coefficients of a given polynomial on ${\mathbb{K}}^{n}$ (${\mathbb{K
}={\mathbb{R}}$ or ${\mathbb{C}}$) has also been widely used in mathematics
and is much easier to handle. Observe that an $m$-homogeneous polynomial in
${\mathbb{K}}^{n}$ can be written as
\[
P(x)={\sum\limits_{\left\vert \alpha\right\vert =m}}a_{\alpha}x^{\alpha},
\]
where $x=(x_{1},\ldots,x_{n})\in{\mathbb{K}}^{n}$, $\alpha=(\alpha_{1
,\ldots,\alpha_{n})\in({\mathbb{N}}\cup\{0\})^{n}$, $|\alpha|=\alpha
_{1}+\cdots+\alpha_{n}$ and $x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n
^{\alpha_{n}}$. Thus we define the $\ell_{p}$ norm of $P$, with $p\geq1$, as
$$
    |P|_p=\left(\sum_{|\alpha|=m}|a_\alpha|^p\right)^\frac{1}{p}.
    $$
If $E$ has finite dimension $n$, then the polynomial norm $\|\cdot\|$ and the
$\ell_{p}$ norm $|\cdot|_{p}$ ($p\geq1$) are equivalent, and therefore there
exist constants $k(m,n), K(m,n)>0$ such that
\begin{equation}
\label{equ:equiv}k(m,n)|P|_{p}\leq\|P\|\leq K(m,n)|P|_{p},
\end{equation}
for all $P\in{\mathcal{P}}(^{m}E)$. The latter inequalities may provide a good
estimate on $\|P\|$ as long as we know the exact value of the best possible
constants $k(m,n)$ and $K(m,n)$ appearing in \eqref{equ:equiv}.

The problem presented above is an extension of the the well known
polynomial Bohnenblust-Hille inequality. It was proved in \cite{bh} that there
exists a constant $D_{m}>0$ such that for every $P\in{\mathcal{P}}(^{m
\ell_{\infty}^{n})$ we have
\begin{equation}
|P|_{\frac{2m}{m+1}}\leq D_{m}\Vert P\Vert.\label{equ:equiv
\end{equation}
Observe that \eqref{equ:equiv} coincides with the first inequality in
\eqref{equ:equiv} for $p=\frac{2m}{m+1}$ except for the fact that $D_{m}$ in
\eqref{equ:equiv} can be chosen in such a way that it is independent from the
dimension $n$. Actually Bohnenblust and Hille showed that $\frac{2m}{m+1}$ is
optimal in \eqref{equ:equiv} in the sense that for $p<\frac{2m}{m+1}$, any
constant $D$ fitting in the inequality
\[
|P|_{p}\leq D\Vert P\Vert,
\]
for all $P\in{\mathcal{P}}(^{m}\ell_{\infty}^{n})$ depends necessarily on $n$.

It was recently shown in \cite{annals2011} that the complex polynomial
Bohnenblust--Hille inequality is hypercontractive. For real scalars, in
\cite{CMPS}, it was proved that the real Bohnenblust--Hille polynomial
inequality is hypercontractive and this result cannot be improved. The
optimality of the result for complex scalars is still open.

The polynomial and multilinear Bohnenblust--Hille inequalities have important applications in different fields of Mathematics and Physics, such as Operator Theory, Fourier and Harmonic Analysis, Complex Analysis, Analytic Number Theory and Quantum Information Theory. Since its origins in 1931, in the \emph{Annals of Mathematics},  the (multilinear and polynomial) Bohnenblust--Hille inequalities were overlooked for a long period (see \cite{bh}) and were only rediscovered in the last few years with works of A. Defant, L. Frerick, J. Ortega-Cerd\'{a}, M. Ouna\"{\i}es, D. Popa, U. Schwarting, K. Seip, among others (see, e.g., \cite{defant2,DD,monta,lama11,addqq,psseo}).


The main motivation of this paper is the following open problem: Is the hypercontractivity of the \emph{complex} polynomial Bohnenblust--Hille inequality an optimal result?

We show that the search of optimal values for $D_{m,{\mathbb{R}}}$ is closely related to the aforementioned problem.

So, we try to determine the value of the best constant in
\eqref{equ:equiv}. This constant depends considerably on whether we consider
the real or complex version of $\ell_{\infty}^{n}$, which motivates the
following definition
\[
D_{m,{\mathbb{K}}}:=\inf\left\{  D>0:\text{$|P|_{\frac{2m}{m+1}}\leq
D_{m}\Vert P\Vert$ for all $P\in{\mathcal{P}}(^{m}\ell_{\infty}^{n})
}\right\}  .
\]


\section{How is the real case connected to the optimality of hypercontractivity of the complex Bohnenblust--Hille inequality?}

In this section we show how the problem of the sharpness of the hypercontractivity of the complex polynomial Bohnenblust--Hille inequality is related to the search of optimal constants for the case of real scalars. We need to introduce an asymptotic approach to the notion of hypercontractivity constant:

\begin{definition}
[Asymptotic approach]The asymptotic hypercontractivity constant of the real
polynomial Bohnenblust--Hille inequality is
\[
H_{\infty,\mathbb{R}}:=\limsup_{m} \sqrt[m]{D_{{\mathbb{R}},m}}.
\]
Observe that for every $\epsilon>0$ there exists $N_{\epsilon}\in{\mathbb{N}}$
such that $D_{{\mathbb{R}},m}\leq(H_{\infty,\mathbb{R}}+\epsilon)^{m}$ for all
$m\geq N_{\epsilon}$. Hence $H_{\infty,\mathbb{R}}$ is a sharp measure of the
asymptotic growth of the constants $D_{{\mathbb{R}},m}$.
\end{definition}


The next result shows the precise connection between the real and complex cases:

\begin{proposition}\label{FF}
If $H_{\infty,\mathbb{R}}>2$ then the hypercontractivity of the complex
polynomial Bohnenblust--Hille inequality is optimal.
\end{proposition}

\begin{proof}
Let $C$ be a real number such tha
\[
H_{\infty,\mathbb{R}}>C>2.
\]
Then there is a sequence $m_{1}<m_{2}<\cdots$ of positive integers so tha
\[
D_{m_{j},\mathbb{R}}>C^{m_{j}
\]
for all $j$. Let $\varepsilon>0$ be a positive integer such tha
\[
C-\varepsilon>2.
\]
Thus, for all such $m_{j}$ there is an $m_{j}$-homogeneous polynomial
\[
P_{m_{j}}={\sum\limits_{\left\vert \alpha\right\vert =m_{j}}}a_{\alpha
}x^{\alpha
\]
so tha
\[
\frac{\left(  \sum_{|\alpha|=m_{j}}|a_{\alpha}|^{\frac{2m_{j}}{m_{j}+1
}\right)  ^{\frac{m_{j}+1}{2m_{j}}}}{\left\Vert P_{m_{j}}\right\Vert }>\left(
C-\varepsilon\right)  ^{m_{j}}.
\]
If $P_{\mathbb{C},m_{j}}$ denotes the complexification of $P_{m_{j}}$ then, from \cite{CMPS} (using a result due to C. Visser \cite{Visser}), we know that
\[
\left\Vert P_{\mathbb{C},m_{j}}\right\Vert \leq2^{m_{j}-1}\left\Vert P_{m_{j
}\right\Vert .
\]
We thus hav
\[
\frac{\left(  \sum_{|\alpha|=m_{j}}|a_{\alpha}|^{\frac{2m_{j}}{m_{j}+1
}\right)  ^{\frac{m_{j}+1}{2m_{j}}}}{\left\Vert P_{\mathbb{C},m_{j
}\right\Vert }\geq\frac{\left(  \sum_{|\alpha|=m_{j}}|a_{\alpha
|^{\frac{2m_{j}}{m_{j}+1}}\right)  ^{\frac{m_{j}+1}{2m_{j}}}}{2^{m_{j
-1}\left\Vert P_{m_{j}}\right\Vert }>\frac{\left(  C-\varepsilon\right)
^{m_{j}}}{2^{m_{j}-1}}>\left(  \frac{C-\varepsilon}{2}\right)  ^{m_{j}}.
\]

\end{proof}

The above result is the motivation of the rest of the paper: to look for lower bounds for the constants of the real polynomial Bohnenblust--Hille inequality.


\section{Lower bounds for the polynomial Bohnenblust-Hille constant}

If $P:\ell_{\infty}^{n}\left(  \mathbb{R}\right)  \rightarrow\mathbb{R}$ is
given by
\[
P(z)={\sum\limits_{\left\vert \alpha\right\vert =m}}a_{\alpha}z^{\alpha},
\]
then
\begin{equation}
D_{\mathbb{R},m}\geq\frac{\left(  {\sum\limits_{\left\vert \alpha\right\vert
=m}}\left|  a_{\alpha}\right|  ^{\frac{2m}{m+1}}\right)  ^{\frac{m+1}{2m}
}{\left\Vert P\right\Vert }=\frac{\left| (a_{\alpha})_{|\alpha|=m}\right|
_{\frac{2m}{m+1}}}{\Vert P\Vert}, \label{vvv
\end{equation}
for all choices of the $a_{\alpha}$'s, where $|\cdot|_{p}$ denotes the usual
$\ell_{p}$ norm. The inequality \eqref{vvv} would provide an easy way to
obtain lower bounds for $D_{\mathbb{R},m}$ if we knew a simple way to compute
$\Vert P\Vert$ for special $P$.

Besides the notion of asymptotic hypercontractivity constant we consider the following alternative approach:

\begin{definition}
[Absolute approach]The hypercontractivity constant of the real polynomial
Bohnenblust--Hille inequality is
\[
H_{a,\mathbb{R}}:=\inf\{H>0:D_{{\mathbb{R}},m}\leq H^{m}\text{ for all $m
}\}.
\]

\end{definition}

We prove that (combining our results with \cite{CMPS}
\[
\left\{
\begin{array}
[c]{c
1.5098<H_{\infty,\mathbb{R}}<2.829\\
1.6561<H_{a,\mathbb{R}}<3.296
\end{array}
\right.
\]
and we provide numerical evidence tha
\[
1.6561<H_{\infty,\mathbb{R}}.
\]


\subsection{The case $m=2$}

Our notation follows \cite{CMPS}. The value of the constant $D_{{\mathbb{R
},2}({\mathcal{P}}(^{2}\ell_{\infty}^{2}))$ can be obtained using the geometry
of the unit ball of ${\mathcal{P}}(\ell_{\infty}^{2})$ described in
\cite{Choi}. We state the result we need for completeness:

\begin{theorem}
\label{the:Choi}[Choi, Kim \cite{Choi}] \label{the:ExtPoints} The extreme
points of the unit ball of ${\mathcal{P}}(^{2}\ell_{\infty}^{2})$ are the
polynomials of the form
\[
\pm x^{2},\ \pm y^{2},\ \pm(tx^{2}-ty^{2}\pm2\sqrt{t(1-t)}xy),
\]
with $t\in[1/2,1]$.
\end{theorem}

As a consequence of the previous result, we obtain the following:

\begin{theorem}
\[
D_{{\mathbb{R}},2}({\mathcal{P}}(^{2}\ell_{\infty}^{2}))=\sup_{t\in
\lbrack1/2,1]} \left[  2t^{\frac{4}{3}}+(2\sqrt{t(1-t)})^{\frac{4}{3}}\right]
^{\frac{3}{4}}  \approx1.8374.
\]
Moreover, the following polynomials are extreme for this problem:
\[
P(x,y)=\pm(0.8678352808x^{2}-0.8678352808y^{2}\pm0.6773395196xy).
\]

\end{theorem}

\begin{proof}
Let
$$f(t)=\left[  2t^{\frac{4}{3}}+(2\sqrt{t(1-t)})^{\frac{4}{3}}\right]
^{\frac{3}{4}}. $$
We just have to notice that due to the convexity of the $\ell_{p}$-norms and
Theorem \ref{the:Choi} we have
\begin{align*}
D_{{\mathbb{R}},2}({\mathcal{P}}(^{2}\ell_{\infty}^{2})) &  =\sup
\{|\mathbf{a}|_{\frac{4}{3}}:\mathbf{a}\in{\mathsf{B}}_{{\mathcal{P}}(^{2
\ell_{\infty}^{2})}\}\\
&  =\sup\{|\mathbf{a}|_{\frac{4}{3}}:\mathbf{a}\in
\ext{\mathsf B}_{{\mathcal{P}}(^{2}\ell_{\infty}^{2})}\}=\sup_{t\in
\lbrack1/2,1]}f(t).
\end{align*}
Observe also that the last supremum is attained at $t\approx0.8678352808$,
concluding the proof.
\end{proof}
\begin{corollary}
\[
D_{{\mathbb{R}},2} \geq 1.8374.
\]
\end{corollary}


\subsection{The case $m=3$}

To the authors' knowledge the calculation of $\|P\|$ is, in general, far from
being easy. However there is a way to compute $\|P\|$ for specific cases. For
instance Grecu, Mu\~{n}oz and Seoane prove in \cite[Lemma 3.12]{GMS} the
following formula:

\begin{lemma}
\label{lem:sup_norm_ab} If for every $a,b\in{\mathbb{R}}$ we define
$P_{a,b}(x,y)=ax^{3}+bx^{2}y+bxy^{2}+ay^{3}$ then
\[
\Vert P_{a,b}\Vert
\begin{cases}
\left\vert a-\frac{b^{2}}{3a}+\frac{2b^{3}}{27a^{2}}+\frac{2a}{27}\left(
-\frac{3b}{a}+\frac{b^{2}}{a^{2}}\right)  ^{\frac{3}{2}}\right\vert  &
\text{if $a\neq0$ and $b_{1}<\frac{b}{a}<3-2\sqrt{3}$},\\
\left\vert 2a+2b\right\vert  & \text{otherwise},
\end{cases}
\]
where
\[
b_{1}=\frac{3}{7}\left(  3-\frac{2\sqrt[3]{9}}{\sqrt[3]{-12+7\sqrt{3}
}+2\sqrt[3]{-36+21\sqrt{3}}\right)  \thickapprox-1.6692.
\]

\end{lemma}

A combination of Lemma \ref{lem:sup_norm_ab} and \eqref{vvv} provide the
following sharp polynomial Bohnenblust-Hille type constant:

\begin{theorem}
Let $P_{a,b}(x,y)=ax^{3}+bx^{2}y+bxy^{2}+ay^{3}$ for $a,b\in{\mathbb{R}}$ and
consider the subset of ${\mathcal{P}}(^{3}\ell_{\infty}^{2})$ given by
$E=\{P_{a,b}:a,b\in{\mathbb{R}}\}$. The
\[
\frac{|(a,b,b,a)|_{\frac{3}{2}}}{\Vert P_{a,b}\Vert}=\left\{
\begin{array}
[c]{c
\frac{27a^{2}\left(  2|a|^{\frac{3}{2}}+2|b|^{\frac{3}{2}}\right)  ^{\frac
{2}{3}}}{\left\vert 27a^{3}-9ab^{2}+2b^{3}+2sign(a)\left(  -3ab+b^{2}\right)
^{\frac{3}{2}}\right\vert },\text{ if $a\neq0$ and $b_{1}<\frac{b}{a
<3-2\sqrt{3}$},\\
\frac{\left(  2|a|^{\frac{3}{2}}+2|b|^{\frac{3}{2}}\right)  ^{\frac{2}{3}
}{2|a+b|},\text{ otherwise
\end{array}
\right.
\]
where $b_{1}$ was defined in Lemma \ref{lem:sup_norm_ab}. Moreover, the above
function attains its maximum when $\frac{b}{a}=b_{1}$, which implies that
\[
D_{{\mathbb{R}},3}(E)=\frac{\left(  2+2|b_{1}|^{\frac{3}{2}}\right)
^{\frac{2}{3}}}{2|1+b_{1}|}\thickapprox2.5525
\]

\end{theorem}

\begin{figure}
\centering
\includegraphics[height=.6\textwidth,keepaspectratio=true]{Pab}
\caption{Graph of the quotient $\frac{|(a,b,b,a)|_\frac{3}{2}}{\|P_{a,b}\|}$ as a function of $\lambda=\frac{a}{b}$.}\label{fig:Pab}
\end{figure}

\begin{corollary}
\[
D_{{\mathbb{R}},3}\geq2.5525.
\]

\end{corollary}
The authors have numerical evidence to state that
\[
D_{{\mathbb{R}},3}({\mathcal{P}}(^{3}\ell_{\infty}^{2}))=D_{{\mathbb{R}
,3}(E),
\]
Moreover, one polynomial for which $D_{{\mathbb{R}},3}({\mathcal{P}}(^{3
\ell_{\infty}^{2}))$ would be attained is approximated by
\[
P_{3}(x,y)=x^{3}-1.6692x^{2}y-1.6692xy^{2}+y^{3}.
\]
One might think that the powers of this polynomial would improve the lower
estimates for the real polynomial Bohnenblust-Hille inequality found in
\cite{CMPS}, but it is not true. Indeed, Notice that
\begin{align*}
P_{3}(x,y)^{4}= &  x^{12}-6.676x^{11}y+10.041x^{10}y^{2}+18.832x^{9
y^{3}-51.359x^{8}y^{4}\\
&  -11.353x^{7}y^{5}+82.242x^{6}y^{6}-11353x^{5}y^{7}-51.359x^{4
y^{8}+18.832x^{3}y^{9}\\
&  +10.041x^{2}y^{10}-6.676xy^{11}+y^{12
\end{align*}
Therefore, if $\mathbf{a}$ is the vector of the coefficients of $P_{3
(x,y)^{4}$ and we use the fact that $\Vert P_{3}\Vert\approx3.2088$, by
\eqref{vvv} we conclude that
\[
D_{{\mathbb{R}},12}\geq\frac{|\mathbf{a}|_{\frac{24}{13}}}{\Vert P_{3
\Vert^{4}}\approx38.1.
\]
The latter estimate is much worse that the estimate $D_{{\mathbb{R}},12
\geq66.39$ found in \cite[Theorem 6.3]{CMPS}.



Despite the powers of $P_{3}$ do not provide an improvement on the lower
estimates obtained on $D_{{\mathbb{R}},m}$, we will see in the next section
that those estimates can be improved by finding $D_{{\mathbb{R}},4}$ numerically.

\subsection{The case $m=4n$}

\begin{theorem}
If $n\in{\mathbb{N}}$, then
\begin{equation}
D_{{\mathbb{R}},4n}\geq D_{{\mathbb{R}},4n}\left(  {\mathcal{P}}(^{4n
\ell_{\infty}^{2})\right)  \geq\frac{\left[  \sum_{k=0}^{n}{\binom{{n}}{{k}
}^{\frac{8n}{4n+1}}\right]  ^{\frac{4n+1}{8n}}}{\left(  {\frac{2\sqrt{3}}{9
}\right)  ^{n}}\geq\frac{\sqrt{(2n)!}}{\left(  {\frac{2\sqrt{3}}{9}}\right)
^{n}n!}.\label{equ:estimate_4
\end{equation}

\end{theorem}

\begin{proof}
Consider the $4$-homogeneous polynomial given by
\[
P_{4}(x,y)=x^{3}y-xy^{3}=xy(x^{2}-y^{2}).
\]
A straightforward calculation shows that $P_{4}$ attains its norm at $\pm
(\pm\frac{1}{\sqrt{3}},1)$ and $\pm(1,\pm\frac{1}{\sqrt{3}})$ and that
$\|P_{4}\|=\frac{2\sqrt{3}}{9}$. Therefore
\[
D_{4}\geq\frac{2^{\frac{5}{8}}}{\frac{2\sqrt{3}}{9}}\approx4.006.
\]
On the other hand $\|P_{4}^{n}\|=\left(  \frac{2\sqrt{3}}{9}\right)  ^{n}$
and
\[
P_{4}(x,y)^{n}=x^{n}y^{n}\sum_{k=0}^{n}{\binom{{n}}{{k}}} x^{2k} y^{2(n-k)}.
\]
Hence, if $\mathbf{a}$ is the vector of the coefficients of $P_{4}$, by
\eqref{vvv} and using the fact that $|\cdot|_{\frac{8n}{4n+1}}\ge|\cdot|_{2}$
(notice that here $|\cdot|_{2}$ is the euclidean norm), we have
\begin{align}
D_{{\mathbb{R}},4n}  &  \geq\frac{|\mathbf{a}|_{\frac{8n}{4n+1}}
{\|P_{4}\|^{n}}=\frac{\left[  \sum_{k=0}^{n}{\binom{{n}}{{k}}}^{\frac
{8n}{4n+1}}\right]  ^{\frac{4n+1}{8n}}}{\left(  {\frac{2\sqrt{3}}{9}}\right)
^{n}}\nonumber\\
&  \ge\frac{\left[  \sum_{k=0}^{n}{\binom{{n}}{{k}}}^{2}\right]  ^{\frac{1
{2}}}{\left(  {\frac{2\sqrt{3}}{9}}\right)  ^{n}}=\frac{\sqrt{\binom{{2n}
{{n}}}}{\left(  {\frac{2\sqrt{3}}{9}}\right)  ^{n}} =\frac{\sqrt{(2n)!
}{\left(  {\frac{2\sqrt{3}}{9}}\right)  ^{n}n!}. \label{ali:lower_estimate
\end{align}
Above we have used the well known formula
\[
\sum_{k=0}^{n}{\binom{{n}}{{k}}}^{2}={\binom{{2n}}{{n}}}.
\]

\end{proof}

\begin{remark}
The following table shows estimates on $D_{{\mathbb{R}},m}$ for $m=4n$,
$n\in{\mathbb{N}}$ based on \eqref{equ:estimate_4}:

\begin{table}[tbh]
\centering
\begin{tabular}
[c]{|l|l||l|l|}\hline
$m=8$ & $D_{\mathbb{R},8} \geq17.4817$ & $m=80$ & $D_{\mathbb{R},80}
\geq7.3769\times10^{13}$\\\hline
$m=12$ & $D_{\mathbb{R},12} \geq81.8865$ & $m=120$ & $D_{\mathbb{R},120}
\geq9.5448\times10^{20}$\\\hline
$m=16$ & $D_{\mathbb{R},16} \geq395.1718$ & $m=160$ & $D_{\mathbb{R},160}
\geq1.2730\times10^{28}$\\\hline
$m=20$ & $D_{\mathbb{R},20} \geq1938.6$ & $m=200$ & $D_{\mathbb{R},200}
\geq1.7261\times10^{35}$\\\hline
$m=24$ & $D_{\mathbb{R},24} \geq9610.8$ & $m=240$ & $D_{\mathbb{R},240}
\geq2.3650\times10^{42}*$\\\hline
$m=28$ & $D_{\mathbb{R},28} \geq4799.2$ & $m=280$ & $D_{\mathbb{R},280}
\geq3.2638\times10^{49}*$\\\hline
$m=32$ & $D_{\mathbb{R},32} \geq2.4093\times10^{5}$ & $m=320$ & $D_{\mathbb{R
,320} \geq4.5279\times10^{56}*$\\\hline
$m=36$ & $D_{\mathbb{R},36} \geq1.2145 \times10^{6}$ & $m=360$ &
$D_{\mathbb{R},360} \geq6.3068\times10^{63}*$\\\hline
$m=40$ & $D_{\mathbb{R},40} \geq6.1418\times10^{6}$ & $m=400$ & $D_{\mathbb{R
,400} \geq8.8123\times10^{70}*$\\\hline
\end{tabular}
\vspace{0.2cm}\caption{Estimates on $D_{\mathbb{R},m}$ for some multiples of
$4$.
\label{tab1
\end{table}Observe that the estimates presented in Table \ref{tab1} certainly
improve those in \cite[Table 1]{CMPS}, however the improvement may not be
asymptotically spectacular since, according to Table \ref{tab1}
$D_{{\mathbb{R}},400}\geq(1.5044)^{400}$, whereas $D_{{\mathbb{R}},400
\geq(1.4896)^{400}$ according to \cite[Table 1]{CMPS}.
\end{remark}

The constant $1.5044$ appearing in the estimate $D_{{\mathbb{R}},400
\geq(1.5044)^{400}$ found in the previous remark is quite accurate as the
following result shows:

\begin{theorem}
If $m=4n$ for $n$ large enough, then
\[
D_{\mathbb{R},m}\geq \left(  {\mathcal{P}}(^{4n
\ell_{\infty}^{2})\right)  \geq\sqrt[4]{\frac{4}{m\pi}}\left(  \sqrt[8
{27}\right)  ^{m}.
\]
Observe that $\sqrt[8]{27}\approx1.5098$.
\end{theorem}

\begin{proof}
Using Stirling's approximation formula $n!\sim\sqrt{2\pi n}\left(  \frac{n
{e}\right)  ^{n}$ in \eqref{ali:lower_estimate} we have, for $m=4n$
\begin{align*}
D_{{\mathbb{R}},4n}\ge\frac{\sqrt{(2n)!}}{\left(  {\frac{2\sqrt{3}}{9
}\right)  ^{n}n!}\sim\frac{\sqrt{2\sqrt{n\pi}\left(  \frac{2n}{e}\right)
^{2n}}}{\left(  {\frac{2\sqrt{3}}{9}}\right)  ^{n}\sqrt{2\pi n}\left(
\frac{n}{e}\right)  ^{n}}=\sqrt[4]{\frac{4}{m\pi}}\left(  \sqrt[8]{27}\right)
^{m}.
\end{align*}

\end{proof}

As a consequence of the latter result and \cite[Theorem 6.4]{CMPS} we have:

\begin{corollary}
\label{cor:H_R}
\[
1.5098<\sqrt[8]{27}\leq H_{\infty,\mathbb{R}}\leq2\sqrt{2}<2.829.
\]
The above result will be improved later.
\end{corollary}

The following corollary, although simple, is stressed because, as it will be
seen later on Theorem \ref{o9}, it highlights a rupture between the real and complex cases.

\begin{corollary}\label{o910}
For all fixed positive integer $n$, the hypercontractivity of the real polynomial Bohnenblust-Hille inequality for
polynomials on ${\mathbb{R}}^{n}$ can not be improved.
\end{corollary}

\subsection{The case $m=5n$}

Let us define the polynomial
\[
P_{5}(x,y)=ax^{5}-bx^{4}y-cx^{3}y^{2}+cx^{2}y^{3}+bxy^{4}-ay^{5},
\]
with
\begin{align*}
a &  =0.194627836350,\\
b &  =0.660089997037,\\
c &  =0.978333058512.
\end{align*}
Using elementary calculus it can be seen that
\[
\Vert P_{5}\Vert=0.286170950359,
\]
up to $10$ decimal places. Then if $\mathbf{a}_{n}$ is the vector of the
coefficients of $P_{5}^{n}$ for each $n\in{\mathbb{N}}$, then we know that
\begin{equation}
D_{{\mathbb{R}},5n}\geq\frac{|\mathbf{a}_{n}|_{\frac{10n}{5n+1}}}{\Vert
P_{5}\Vert^{n}},\label{equ:case_5
\end{equation}
in particular one can check numerically that
\[
D_{{\mathbb{R}},5}\geq6.835918785877.
\]
The authors have numerical evidence showing that $D_{{\mathbb{R}
,5}({\mathcal{P}}(^{5}\ell_{\infty}^{2}))=6.8359$ up to 4 decimal places.

A table with estimates on $D_{{\mathbb{R}},5n}$ obtained by using
\eqref{equ:case_5} for different values of $n\in{\mathbb{N}}$ can be found in
Table \ref{tab2}.

\begin{table}[tbh]
\centering
\begin{tabular}
[c]{|l|l||l|l|}\hline
$m=10$ & $D_{\mathbb{R},10} \geq48.03065$ & $m=80$ & $D_{\mathbb{R},80
\geq9.90603\times10^{11}$\\\hline
$m=15$ & $D_{\mathbb{R},15} \geq399.007$ & $m=120$ & $D_{\mathbb{R},120}
\geq2.83620\times10^{22}$\\\hline
$m=20$ & $D_{\mathbb{R},20} \geq3271.54$ & $m=160$ & $D_{\mathbb{R},160
\geq1.19496\times10^{30}$\\\hline
$m=25$ & $D_{\mathbb{R},25} \geq28308.7$ & $m=200$ & $D_{\mathbb{R},200
\geq5.11958\times10^{37}$\\\hline
$m=30$ & $D_{\mathbb{R},30} \geq2.41034\times10^{5}$ & $m=240$ &
$D_{\mathbb{R},240}\geq2.21659\times10^{45}$\\\hline
$m=35$ & $D_{\mathbb{R},35} \geq2.11695\times10^{6}$ & $m=280$ &
$D_{\mathbb{R},280}\geq9.66672 \times10^{52}$\\\hline
$m=40$ & $D_{\mathbb{R},40} \geq1.83355\times10^{7}$ & $m=320$ &
$D_{\mathbb{R},320} \geq4.23805 \times10^{59}$\\\hline
$m=45$ & $D_{\mathbb{R},45} \geq1.62275\times10^{8}$ & $m=360$ &
$D_{\mathbb{R},360} \geq1.86553\times10^{68}$\\\hline
$m=50$ & $D_{\mathbb{R},50} \geq1.41925\times10^{9}$ & $m=400$ &
$D_{\mathbb{R},400} \geq8.23785\times10^{75}$\\\hline
\end{tabular}
\vspace{0.2cm}\caption{Estimates on $D_{\mathbb{R},m}$ for some multiples of
$5$.
\label{tab2
\end{table}

\begin{remark}
Observe that using the estimates in Table \ref{tab2} we obtain
\[
D_{\mathbb{R},400}\geq(1.5480)^{400},
\]
whereas using the estimates in table \ref{tab1} we have
\[
D_{\mathbb{R},400}\geq(1.5044)^{400}.
\]
It is interesting to notice too that $\sqrt[8]{27}\approx1.5098<1.5491$, where
is the lower bound on $H_{\mathbb{R}}$ found in Corollary \ref{cor:H_R}.
Hence, the results that have appeared in this section suggest that we have the
following:
\begin{equation}
H_{\infty,\mathbb{R}}\geq1.5480. \label{hyy
\end{equation}

\end{remark}

Concerning the absolute hypercontractivity constant we have (combining our
results with \cite{CMPS}):

\begin{corollary
\[
1.5480\leq H_{a,\mathbb{R}}<3.296
\]
\bigskip
\end{corollary}

\subsection{The case $m=6n$}

The authors have numerical evidence pointing to the fact that the extreme
polynomial in the Bohnenblust-Hille inequality for polynomials in
${\mathcal{P}}(^{6}\ell_{\infty}^{2})$ is of the form
\[
Q_{a,b}(x,y)=ax^{5}y+bx^{3}y^{3}+axy^{5}.
\]
This motivates a deeper study of this type of polynomials, which we do in the
following result.

\begin{theorem}
Let $Q_{a,b}(x,y)=ax^5y+bx^3y^3+axy^5$ for $a,b\in{\mathbb{R}}$ and
consider the subspace of ${\mathcal{P}}(^{6}\ell_{\infty}^{2})$ given by
$F=\{Q_{a,b}:a,b\in{\mathbb{R}}\}$. Suppose $\lambda_0<\lambda_1$ are the only two roots of the equation
    $$
    \frac{|3\lambda^2-20+\lambda\sqrt{9\lambda^2-20}|}{25}\sqrt{\frac{-3\lambda-\sqrt{9\lambda^2-20}}{10}}=|2+\lambda|.
    $$
Then if $\lambda =\frac{b}{a}$ we have

\[
\frac{|(0,a,0,b,0,a,0)|_{\frac{12}{7}}}{\Vert Q_{a,b}\Vert}=\left\{
\begin{array}
[c]{c
\frac{25\sqrt{10}\left(2+|\lambda|^{\frac{12}{7}}\right)^{\frac
{12}{7}}}{\left|3\lambda^2-20+\lambda\sqrt{9\lambda^2-20}\right|\sqrt{-3\lambda-\sqrt{9\lambda^2-20}} },\text{ if $a\neq0$ and $\lambda_0<\frac{b}{a
<\lambda_1$},\\
\frac{\left(  2+|\lambda|^{\frac{12}{7}}\right)^{\frac{7}{12}}
}{|2+\lambda|},\text{ otherwise.
\end{array}
\right.
\]
Observe that $\lambda_0\approx -2.2654$, $\lambda_1\approx -1.6779$ and the above
function attains its maximum when $\frac{b}{a}=\lambda_0$ (see Figure \ref{fig:Qab}), which implies that
\[
D_{{\mathbb{R}},6}(F)=\frac{\left(  2+|\lambda_{0}|^{\frac{12}{7}}\right)
^{\frac{12}{7}}}{|2+\lambda_{0}|}\thickapprox 10.7809.
\]
\end{theorem}

\begin{proof}
We do not lose generality by considering only polynomials of the form $Q_{1,\lambda}$, in which case
    $$
    \|Q_{1,\lambda}\|=\sup\{|x^5+\lambda x^3+x|:x\in[0,1]\}.
    $$
The polynomial $q_{\lambda}(x):=x^5+\lambda x^3+x$ has no critical points if $\lambda>-\frac{2\sqrt{5}}{3}$, otherwise it has the following critical points in $[0,1]$:
    $$
        x_0:=\sqrt{\frac{-3\lambda-\sqrt{9\lambda^2-20}}{10}}\quad\text{and}\quad
        x_1:=\sqrt{\frac{-3\lambda+\sqrt{9\lambda^2-20}}{10}}\quad\text{if $-2\leq \lambda\leq -\frac{2\sqrt{5}}{3}$,}
    $$
and $x_0$ if $\lambda\leq -2$. Notice that
    \begin{align*}
     q_\lambda(x_0)&= \frac{-3\lambda^2+20-\lambda\sqrt{9\lambda^2-20}}{20}x_0\\
     q_\lambda(x_1)&= \frac{-3\lambda^2+20+\lambda\sqrt{9\lambda^2-20}}{20}x_1.
    \end{align*}
It is easy to check that $|q_\lambda(x_0)|\geq |q_\lambda(x_1)|$ for $-2\leq \lambda\leq -\frac{2\sqrt{5}}{3}$, which implies that
    $$
    \|Q_{1,\lambda}\|=\begin{cases}
    \max\{|2+\lambda|,|q_\lambda(x_0)|\}&\text{if $-2\leq \lambda\leq -\frac{2\sqrt{5}}{3}$,}\\
    |2+\lambda|&\text{otherwise.}
    \end{cases}
    $$
The equation $|2+\lambda|=|q_\lambda(x_0)|$ turns out to have only two roots, namely $\lambda_0\approx -2.2654$ and $\lambda_1\approx -1.6779$. By continuity, it is easy to prove that $|2+\lambda|\leq |q_\lambda(x_0)|$ only if $-2\leq \lambda\leq -\frac{2\sqrt{5}}{3}$, which concludes the proof.
\end{proof}

\begin{corollary}
\[
D_{{\mathbb{R}},6}\geq10.7809.
\]
\end{corollary}

\begin{figure}
\centering
\includegraphics[height=.6\textwidth,keepaspectratio=true]{Qab}
\caption{Graph of the quotient $\frac{|(0,a,0,b,0,a,0)|_\frac{12}{7}}{\|Q_{a,b}\|}$ as a function of $\lambda=\frac{b}{a}$.}\label{fig:Qab}
\end{figure}

As we did in the previous cases, it would be interesting to know if we can
improve our best lower bound on $D_{{\mathbb{R}},m}$ by considering powers of

\[
P_{6}(x,y)=Q_{1,\lambda_{0}}(x,y)=x^{5}y+\lambda_{0}x^{3}y^{3}+xy^{5},
\]
with $\lambda_{0}=-2.2654$.

If $\mathbf{a}_{n}$ is the vector of the coefficients of $P_{6}^{n}$ for each
$n\in{\mathbb{N}}$, then we know that
\begin{equation}
\label{equ:case_6}D_{{\mathbb{R}},6n}\geq\frac{|\mathbf{a}_{n}|_{\frac
{12n}{6n+1}}}{\|P_{6}\|^{n}}.
\end{equation}


Using \eqref{equ:case_6} we can construct a table (see Table \ref{tab3} of
lower estimates for $D_{{\mathbb{R}},6m}$ that improves the results obtained
in tables \ref{tab1} and \ref{tab2}.

\begin{table}[tbh]
\centering
\begin{tabular}
[c]{|l|l||l|l|}\hline
$m=12$ & $D_{\mathbb{R},12} \geq144.057$ & $m=120$ & $D_{\mathbb{R},120
\geq4.02749\times10^{23}$\\\hline
$m=18$ & $D_{\mathbb{R},18} \geq2078.73$ & $m=150$ & $D_{\mathbb{R},150}
\geq4.07526\times10^{29}$\\\hline
$m=24$ & $D_{\mathbb{R},24} \geq30958.8$ & $m=180$ & $D_{\mathbb{R},180
\geq4.16733\times10^{35}$\\\hline
$m=30$ & $D_{\mathbb{R},30} \geq4.69119\times10^{5}$ & $m=210$ &
$D_{\mathbb{R},210}\geq4.29250\times10^{41}$\\\hline
$m=36$ & $D_{\mathbb{R},36} \geq7.18661\times10^{6}$ & $m=240$ &
$D_{\mathbb{R},240}\geq4.44489\times10^{47}$\\\hline
$m=42$ & $D_{\mathbb{R},42} \geq1.10924\times10^{8}$ & $m=270$ &
$D_{\mathbb{R},270}\geq4.62131\times10^{53}$\\\hline
$m=48$ & $D_{\mathbb{R},48} \geq1.72150\times10^{9}$ & $m=300$ &
$D_{\mathbb{R},300} \geq4.82001\times10^{59}$\\\hline
$m=54$ & $D_{\mathbb{R},54} \geq2.68289\times10^{10}$ & $m=360$ &
$D_{\mathbb{R},360} \geq5.28156\times10^{71}$\\\hline
$m=60$ & $D_{\mathbb{R},60} \geq4.19492\times10^{11}$ & $m=420$ &
$D_{\mathbb{R},420} \geq5.82897\times10^{83}$\\\hline
\end{tabular}
\vspace{0.2cm}\caption{Estimates on $D_{\mathbb{R},m}$ for some multiples of
$6$.
\label{tab3
\end{table}

\begin{remark}
Observe that using the estimates in Table \ref{tab3} we obtain
\[
D_{\mathbb{R},420}\geq(1.5828)^{420},
\]
whereas using the polynomial $P_{5}$ as in the construction of Table \ref{tab2} we
would have
\[
D_{\mathbb{R},420}\geq(1.5483)^{420}.
\]
The last inequality suggests that
\begin{equation}
H_{\infty,\mathbb{R}}\geq1.5828. \label{hyYy
\end{equation}

\end{remark}

Concerning the absolute hypercontractivity constant we have (combining our
results with \cite{CMPS}):

\begin{corollary
\[
1.5828\leq H_{a,\mathbb{R}}<3.296
\]
\bigskip
\end{corollary}

\subsection{The case $m=2^{n}$ and $n>1$}

For $m=4$ let us consider
\[
P_{4}(x_{1},...,x_{4})=P_{2}(x_{1},x_{2})^{2}-P_{2}(x_{3},x_{4})^{2}=\left(
x_{1}^{2}-x_{2}^{2}\right)  ^{2}-\left(  x_{3}^{2}-x_{4}^{2}\right)  ^{2}.
\]
We thus have
\[
D_{4}\geq\frac{\left(  4+2\times2^{8/5}\right)  ^{5/8}}{1}\geq4.2335=\left(
1.4344\right)  ^{4
\]
Using induction we conside
\[
P_{2^{n}}(x_{1},...,x_{2^{n}})=P_{2^{n-1}}(x_{1},...,x_{2^{n-1}
)^{2}-P_{2^{n-1}}(x_{2^{n-1}+1},...x_{2^{n}})^{2}.
\]
For $n=3,4,5$ we were able to comput
\begin{align*}
D_{8}  &  \geq\left(  1.5241\right)  ^{8}\\
D_{16}  &  \geq\left(  1.59527998\right)  ^{16}\\
D_{32}  &  \geq\left(  1.65617484\right)  ^{32
\end{align*}
and these results are quite better than those from the previous sections when
$m=2^{n}$. We thus conclude tha
\[
H_{a,\mathbb{R}}\geq1.65617484
\]
and we have numerical evidence that in fact (\ref{hyy}) can be improved to
\[
H_{\infty,\mathbb{R}}\geq1.65617484.
\]


The following table summarizes our estimates for special values of $m:$

\[
\begin{tabular}[c]{|c|c|}\hline
$m$ & $D_{\mathbb{R},m}\geq$\\
\hline
$2$ & $1.8374=\left(  1.3555\right)  ^{2}$\\
\hline
$3$ & $2.5525=\left(  1.3666\right)  ^{3}$\\
\hline
$4$ & $4.2335=\left(  1.4344\right)  ^{4}$\\
\hline
$5$ & $6.8359=\left(  1.4688\right)  ^{5}$\\
\hline
$6$ & $10.7809=\left(  1.4863\right)  ^{6}$\\
\hline
$8$ & $29.1209=\left(  1.5241\right)  ^{8}$\\
\hline
$16$ & $1.59527998^{16}$\\
\hline
$32$ & $1.65617484^{32}$\\
\hline
\end{tabular}
\]
The above table induces us to believe that it is quite likely that by
computing $D_{m}$ for $m=2^{n}$ and big values of $m$ we will substantially
improve the value $1.65617484.$

\section{Variants of the polynomial Bohnenblust-Hille inequalities}

\subsection{A result for both complex and real scalars}

The optimality of the power $\frac{2m}{m+1}$ means that for $r<\frac{2m}{m+1}$
the factor $D_{\mathbb{C},m}$ will appear multiplied by a factor depending on
$n$. The next result shows that this factor is precisely $n^{\left(  \frac
{m}{r}-\frac{m+1}{2}\right)  }.$ The power $\frac{2m}{m+1}$ is the precise
value where the dependence on $n$ disappears:

\begin{proposition}
\label{pol11}Let $r\in\lbrack1,\frac{2m}{m+1}],$ $m,n$ be positive integers
and $\mathbb{K}=$ $\mathbb{R}$ or $\mathbb{C}.$ There is an universal constant
$L_{\mathbb{K}}>0$ such tha
\begin{equation}
\left\Vert P\right\Vert _{r}\leq L_{\mathbb{K}}^{m}\cdot n^{\left(  \frac
{m}{r}-\frac{m+1}{2}\right)  }\left\Vert P\right\Vert _{\infty} \label{ki
\end{equation}
for all $m$-homogeneous polynomials $P:\ell_{\infty}^{n}\left(  \mathbb{K
\right)  \rightarrow\mathbb{K}.$ Moreover, the power $\frac{m}{r}-\frac
{m+1}{2}$ is optimal.
\end{proposition}

\begin{proof}
Let $r\in$ $[1,\frac{2m}{m+1}].$ The proof of (\ref{ki}) for complex scalars
is easily obtained by using the Holder Inequality in combination with the case
$\frac{2m}{m+1};$ we learned this argument from \cite{fr}. For real scalars,
according to \cite{CMPS}, if $P:\ell_{\infty}^{n}\left(  \mathbb{R}\right)
\rightarrow\mathbb{R}$ is an $m$-homogeneous polynomial, then (as in Proposition \ref{FF}),
\[
\left\Vert P_{\mathbb{C}}\right\Vert _{\infty}\leq2^{m-1}\left\Vert
P\right\Vert _{\infty},
\]
where $P_{\mathbb{C}}$ is the complexification of $P$. So, we obtain (\ref{ki}) for real
scalars. It also simple to see that the constant $L_{\mathbb{K}}$ can be
chosen independent of $m,r,n.$ Now let us prove the optimality of the exponent
$\frac{m}{r}-\frac{m+1}{2};$ for this task let us suppose that the result
holds for a power $q<\frac{m}{r}-\frac{m+1}{2}$.

For each $m,n$, let
\begin{align*}
P_{m,n}  &  :\ell_{\infty}^{n}\left(  \mathbb{K}\right)  \rightarrow
\mathbb{K}\\
P_{m,n}(w)  &  ={\sum\limits_{\left\vert \alpha\right\vert =m}}\varepsilon
_{\alpha}w^{\alpha
\end{align*}
be the $m$-homogeneous Bernoulli polynomial satisfying the
Kahane--Salem--Zygmund inequality (note that this inequality is also valid for
real scalars, see \cite{Nun2}).

The proof follows the lines of \cite[Theorem 10.2]{Nun2}; the essence of this
argument can be traced back to Boas' classical paper \cite{Boas}. We can
suppose $n>m.$ As in \cite{Nun2}, we have
\[
{\textstyle\sum_{\left\vert \alpha\right\vert =m}}\left\vert \varepsilon
_{\alpha}\right\vert ^{r}=p(n)+\frac{1}{m!}{\textstyle\prod\limits_{k=0
^{m-1}}(n-k),
\]
where $p\left(  n\right)  >0$ is a polynomial of degree $m-1.$ If (\ref{ki})
was valid with the power $q$, then there would exist a constant
$C_{q,\mathbb{K}}>0$ so tha
\begin{align*}
\left(  {\sum_{\left\vert \alpha\right\vert =m}}\left\vert \varepsilon
_{\alpha}\right\vert ^{r}\right)  ^{\frac{1}{r}}  &  \leq C_{q,\mathbb{K}
^{m}\cdot n^{q}\left\Vert P_{m,n}\right\Vert _{\infty}\\
&  \leq C_{q,\mathbb{K}}^{m}\cdot n^{q}\cdot C_{KSZ}\cdot n^{\left(
m+1\right)  /2}\sqrt{\log m},
\end{align*}
where $C_{KSZ}>0$ is the universal constant from the Kahane--Salem--Zygmund
inequality. Henc
\[
C_{q,\mathbb{K}}^{m}C_{KSZ}\geq\frac{1}{n^{q}\cdot n^{\left(  m+1\right)
/2}\sqrt{\log m}}\left(  p(n)+\frac{1}{m!}{\textstyle\prod\limits_{k=0}^{m-1
}(n-k)\right)  ^{1/r
\]
for all $n$. Raising both sides to the power of $r$ and letting $n\rightarrow
\infty$ we obtai
\[
\left(  C_{q,\mathbb{K}}^{m}C_{KSZ}\right)  ^{r}\geq\lim_{n\rightarrow\infty
}\left(  \frac{p(n)}{n^{qr}\cdot n^{r\left(  m+1\right)  /2}\left(  \sqrt{\log
m}\right)  ^{r}}+\frac{s(n)}{n^{qr}\cdot n^{r\left(  m+1\right)  /2}\left(
\sqrt{\log m}\right)  ^{r}}\right)  ,
\]
with
\[
s(n)=\frac{1}{m!}{\textstyle\prod\limits_{k=0}^{m-1}}(n-k).
\]
Since $q<\frac{m}{r}-\frac{m+1}{2}$, we have $\deg s=m>qr+r(m+1)/2$ and thus
the limit above is infinity, a contradiction.
\end{proof}

\subsection{Another rupture between the cases of real and complex scalars}

If we replace $\frac{2m}{m+1}$ by $q>\frac{2m}{m+1}$ in the polynomial
Bohnenblust--Hille inequality it is natural to investigate if, at some point
on, the dependence on the factor depending on $m$ disappears. More precisely,
we conside
\[
\rho_{\mathbb{K}}=\inf\left\{  q;\left\Vert P\right\Vert _{q}\leq\left\Vert
P\right\Vert _{\infty}\text{ for all }m\text{-homogeneous }P:\ell_{\infty
^{n}\left(  \mathbb{K}\right)  \rightarrow\mathbb{K}\text{ }\right\}  .
\]
For $q=2$ and complex scalars, it is well-known (see \cite{seip3} and details can be found in the proof of the Theorem \ref{o9}) that
\[
\left\Vert P\right\Vert _{2}\leq\left\Vert P\right\Vert _{\infty
\]
for all $m$ and $n$, and thu
\[
\rho_{\mathbb{C}}\leq2.
\]


From a recent result of D. Nu\~{n}ez-Alarc\'{o}n (\cite{Nu3}), we know that
for all $m$ there is an $m$-homogeneous polynomial so tha
\[
\left\Vert P\right\Vert _{\frac{2m}{m+1}}>\left\Vert P\right\Vert _{\infty}.
\]
More precisely
\[
D_{\mathbb{C},m}\geq\frac{\left(  2+2^{m}\right)  ^{\frac{m+1}{2m}}
{\sqrt{4+2^{m+1}}}>1.
\]
We thus have:

\begin{proposition}
$\rho_{\mathbb{C}}=2.$
\end{proposition}

For real scalars, however, from the previous section it is easy to see tha
\[
\rho_{\mathbb{R}}=\infty.
\]


\section{Bohnenblust--Hille type inequalities and multilinear forms}

The multilinear Bohnenblust--Hille inequality (1931, \cite{bh}) asserts that
for every positive integer $m\geq2$ there exists a sequence of positive
scalars $C_{\mathbb{C},m}\geq1$ such that
\begin{equation}
\left(  \sum\limits_{i_{1},\ldots,i_{m}=1}^{N}\left\vert U(e_{i_{^{1}}},\ldots,e_{i_{m}})\right\vert ^{\frac{2m}{m+1}}\right)  ^{\frac{m+1}{2m}}\leq
C_{\mathbb{C},m}\sup_{z_{1},...,z_{m}\in\mathbb{D}^{N}}\left\vert U(z_{1},...,z_{m})\right\vert \label{hypp}
\end{equation}
for all $m$-linear mapping $U:\mathbb{C}^{N}\times\cdots\times\mathbb{C}^{N}\rightarrow\mathbb{C}$ and every positive integer $N$, where $\left(
e_{i}\right)  _{i=1}^{N}$ denotes the canonical basis of $\mathbb{C}^{N}.$

The result is also valid for real scalars and $\frac{2m}{m+1}$ is optimal in both real and complex cases. As in the polynomial case, the only differences
between the complex and real scalars appear in the best constants involved. For example
\[
C_{\mathbb{C},2}\leq\frac{2}{\sqrt{\pi}}<\sqrt{2}=C_{\mathbb{R},2}.
\]

\subsection{A variant for both complex and real scalars}

The next proposition is the multilinear version of Theorem \ref{pol11}.

\begin{proposition}
Let $r\in\lbrack1,\frac{2m}{m+1}],$ $m,n$ be positive integers and
$\mathbb{K}=$ $\mathbb{R}$ or $\mathbb{C}.$ There is an universal constant
$C_{m}>0$ such that
\[
\left\Vert T\right\Vert _{r}\leq C_{m}\cdot n^{\left(  \frac{m}{r}-\frac
{m+1}{2}\right)  }\left\Vert T\right\Vert _{\infty}
\]
for all $m$-linear forms $T:\ell_{\infty}^{n}\left(  \mathbb{K}\right)
\times\cdots\times\ell_{\infty}^{n}\left(  \mathbb{K}\right)  \rightarrow
\mathbb{K}.$ Moreover, the power $\frac{m}{r}-\frac{m+1}{2}$ is optimal.
\end{proposition}

\begin{proof}
Let
\[
\left\Vert T\right\Vert _{r}:=\left(  {\textstyle}\sum_{i_{1},...,i_{m}=1
^{n}\left\vert T\left(  e_{i_{1}},...,e_{i_{m}}\right)  \right\vert
^{r}\right)  ^{1/r}.
\]
Then, for $1\leq r<\frac{2m}{m+1},$
\begin{align*}
&  \left(  {\textstyle\sum_{i_{1},...,i_{m}=1}^{n}}\left\vert T\left(
e_{i_{1}},...,e_{i_{m}}\right)  \right\vert ^{r}\right)  \\
&  \leq\left(  \left(  {\textstyle\sum_{i_{1},...,i_{m}=1}^{n}}\left\vert
1\right\vert ^{r}\right)  ^{\frac{2m}{2m-rm-r}}\right)  ^{\frac{2m-rm-r}{2m
}\left(  {\textstyle\sum_{i_{1},...,i_{m}=1}^{n}}\left(  \left\vert T\left(
e_{i_{1}},...,e_{i_{m}}\right)  \right\vert ^{r}\right)  ^{\frac{2m}{r\left(
m+1\right)  }}\right)  ^{\frac{r\left(  m+1\right)  }{2m}}\\
&  \leq\left(  n^{m}\right)  ^{\frac{2m-rm-r}{2m}}\left(  {\textstyle\sum
_{i_{1},...,i_{m}=1}^{n}}\left\vert T\left(  e_{i_{1}},...,e_{i_{m}}\right)
\right\vert ^{\frac{2m}{\left(  m+1\right)  }}\right)  ^{\frac{r\left(
m+1\right)  }{2m}}\\
&  =n^{\frac{2m-rm-r}{2}}\left(  {\textstyle\sum_{i_{1},...,i_{m}=1}^{n
}\left\vert T\left(  e_{i_{1}},...,e_{i_{m}}\right)  \right\vert ^{\frac
{2m}{\left(  m+1\right)  }}\right)  ^{\frac{r\left(  m+1\right)  }{2m}
\end{align*}
and thus
\[
\left(  {\textstyle\sum_{i_{1},...,i_{m}=1}^{n}}\left\vert T\left(  e_{i_{1
},...,e_{i_{m}}\right)  \right\vert ^{r}\right)  ^{1/r}\leq n^{\frac{m
{r}-\frac{\left(  m+1\right)  }{2}}C_{m}\left\Vert T\right\Vert .
\]
Let $P:\ell_{\infty}^{n}(\mathbb{K})\rightarrow\mathbb{K}$,
\[
P(z)={\sum\limits_{\left\vert \alpha\right\vert =m}}a_{\alpha}z^{\alpha},
\]
be an $m$-homogeneous polynomial and $T$ be its polar. Then
\[
{\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha
}\right\vert ^{r}={\sum\limits_{\left\vert \alpha\right\vert =m}}\binom
{m}{\alpha}^{r}\left\vert T(e_{1}^{\alpha_{1}},\ldots,e_{n}^{\alpha_{n
})\right\vert ^{r}.
\]
However, for every choice of $\alpha$, the term
\[
\left\vert T(e_{1}^{\alpha_{1}},\ldots,e_{n}^{\alpha_{n}})\right\vert ^{r
\]
is repeated $\binom{m}{\alpha}$ times in the sum
\[
{\sum\limits_{i_{1},\ldots,i_{m}=1}^{n}}\left\vert T(e_{i_{1}},\ldots
,e_{i_{m}})\right\vert ^{r}.
\]
Thus
\[
{\sum\limits_{\left\vert \alpha\right\vert =m}}\binom{m}{\alpha}^{r}\left\vert
T(e_{1}^{\alpha_{1}},\ldots,e_{n}^{\alpha_{n}})\right\vert ^{r}={\sum
\limits_{i_{1},\ldots,i_{m}=1}^{n}}\binom{m}{\alpha}^{r}\frac{1}{\binom
{m}{\alpha}}\left\vert T(e_{i_{1}},\ldots,e_{i_{m}})\right\vert ^{r
\]
and, since
\[
\binom{m}{\alpha}\leq m!
\]
we have
\[
{\sum\limits_{\left\vert \alpha\right\vert =m}}\binom{m}{\alpha}^{r}\left\vert
T(e_{1}^{\alpha_{1}},\ldots,e_{n}^{\alpha_{n}})\right\vert ^{r}\leq\left(
m!\right)  ^{r-1}{\sum\limits_{i_{1},\ldots,i_{m}=1}^{n}}\left\vert
T(e_{i_{1}},\ldots,e_{i_{m}})\right\vert ^{r}.
\]
We finally obtain
\begin{align*}
\left(  {\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha
}\right\vert ^{r}\right)  ^{r} &  \leq\left(  \left(  m!\right)  ^{r-1
{\sum\limits_{i_{1},\ldots,i_{m}=1}^{n}}\left\vert T(e_{i_{1}},\ldots
,e_{i_{m}})\right\vert ^{r}\right)  ^{\frac{1}{r}}\\
&  \leq\left(  m!\right)  ^{\frac{r-1}{r}}n^{\frac{m}{r}-\frac{\left(
m+1\right)  }{2}}.C_{m}\left\Vert T\right\Vert .
\end{align*}
On the other hand, it is well-known tha
\[
\left\Vert T\right\Vert \leq\frac{m^{m}}{m!}\left\Vert P\right\Vert
\]
and hence
\begin{align*}
\left(  {\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha
}\right\vert ^{r}\right)  ^{1/r} &  \leq\left(  m!\right)  ^{\frac{r-1}{r
}n^{\frac{m}{r}-\frac{\left(  m+1\right)  }{2}}C_{m}\frac{m^{m}}{m!}\left\Vert
P\right\Vert \\
&  =C_{m}\frac{m^{m}}{\left(  m!\right)  ^{1-\frac{r-1}{r}}}n^{\frac{m
{r}-\frac{\left(  m+1\right)  }{2}}\left\Vert P\right\Vert \\
&  =C_{m}\frac{m^{m}}{\left(  m!\right)  ^{1/r}}n^{\frac{m}{r}-\frac{\left(
m+1\right)  }{2}}\left\Vert P\right\Vert .
\end{align*}
Now the Kahane--Salem--Zygmund as in the proof of Theorem \ref{pol11} asserts
that $\frac{m}{r}-\frac{\left(  m+1\right)  }{2}$\ is optimal.

\end{proof}



\subsection{Other parameters}

Now let us define
\[
\mu_{\mathbb{K}}=\inf\left\{  q;\left\Vert T\right\Vert _{q}\leq\left\Vert
T\right\Vert _{\infty}\text{ for all }m\text{-linear form }T:\ell_{\infty
^{n}\left(  \mathbb{K}\right)  \times\cdots\times\ell_{\infty}^{n}\left(
\mathbb{K}\right)  \rightarrow\mathbb{K}\right\}  .
\]
For both real and complex scalars it is well-known tha
\[
\left\Vert T\right\Vert _{2}\leq\left\Vert T\right\Vert _{\infty
\]
for all $T:\ell_{\infty}^{n}\left(  \mathbb{K}\right)  \times\cdots\times
\ell_{\infty}^{n}\left(  \mathbb{K}\right)  \rightarrow\mathbb{K}.$ Henc
\[
\mu_{\mathbb{K}}\leq2.
\]
However no more information is available for the complex case. In fact no
nontrivial lower bounds for the constants of the complex multilinear
Bohnenblust--Hille are known. All we know is tha
\[
C_{\mathbb{C},m}\geq1.
\]
\ For real scalars we have:

\begin{proposition}
$\mu_{\mathbb{R}}=2.$
\end{proposition}

\begin{proof}
Let $q<2.$ As in (\cite{diniz2}), considering $T_{2}:\ell_{\infty}^{n}\left(
\mathbb{R}\right)  \times\ell_{\infty}^{n}\left(  \mathbb{R}\right)
\rightarrow\mathbb{R}$ given by
\[
T_{2}(x,y)=x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2
\]
we hav
\[
\left\Vert T_{2}\right\Vert =2
\]
and i
\[
\left(  \sum\limits_{i_{1},i_{2}=1}^{2}\left\vert T_{2}(e_{i_{^{1}}},e_{i_{2
})\right\vert ^{q}\right)  ^{\frac{1}{q}}\leq C_{2}\left\Vert T_{2
\right\Vert
\]
we get
\[
C_{2}\geq2^{\frac{2}{q}-1}>1.
\]
Following the same induction process from (\cite{diniz2}), we hav
\[
\left(  4^{m-1}\right)  ^{\frac{1}{q}}\leq C_{m}2^{m-1
\]
and thu
\[
C_{m}\geq2^{\frac{m\left(  2-q\right)  -\left(  2-q\right)  }{q}}>1.
\]
\bigskip
\end{proof}



\section{Contractivity in finite dimensions}


In the present section we will prove that the complex polynomial
Bohnenblust-Hille constants for polynomials on ${\mathbb{C}}^{n}$, with $n\in{\mathbb N}$ fixed, are
contractive, and not hypercontractive as it happens for real polynomials on
${\mathbb{R}}^{n}$ (Corollary \ref{o910}). Therefore, if we want to prove the sharpness of the hypercontractivity of
the complex polynomial Bohnenblust-Hille constants we have to search for
polynomials in growing number of variables.

\begin{theorem}
\label{o9} For all $n\geq2$ the complex polynomial Bohnenblust--Hille
inequality is contractive in $\mathcal{P}(^{m}\ell_{\infty}^{n}).$ More
precisely, for all fixed $n\in{\mathbb{N}}$, there are constants $D_{m},$ with
$\lim_{m\rightarrow\infty}D_{m}=1$, so that
\[
|P|_{\frac{2m}{m+1}}\leq D_{m}\Vert P\Vert
\]
for all $P\in\mathcal{P}(^{m}\ell_{\infty}^{n}).$
\end{theorem}

\begin{proof}
Let $P(z)=\sum_{|\alpha|=m}c_{\alpha}z^{\alpha}$ and $f(t)=P(e^{it_{1}
,\ldots,e^{it_{n}})=\sum_{|\alpha|=m}c_{\alpha}e^{i\alpha t}$, where
$t=(t_{1},\ldots,t_{n})\in{\mathbb{R}}^{n}$ $\alpha\in({\mathbb{N}
\cup\{0\})^{n}$ and $\alpha t=\alpha_{1}t_{1}+\cdots+\alpha_{n}t_{n}$. .
Observe that if $\Vert f\Vert$ denotes the sup norm of $f$ on $[-\pi,\pi]$, by
the Maximum Modulus Principle $\Vert f\Vert=\Vert P\Vert$. Also, due to the
orthogonality of the system $\{e^{iks}:n\in{\mathbb{Z}}\}$ in $L^{2}([-\pi
,\pi])$ we have
\[
\Vert P\Vert^{2}=\Vert f\Vert^{2}\geq\frac{1}{2\pi}\int_{-\pi}^{\pi
|f(t)|^{2}dt=\sum_{|\alpha|=m}|c_{\alpha}|^{2}=|P|_{2}^{2},
\]
from which $|P|_{2}\leq\Vert P\Vert$. On the other hand it is well known that
in ${\mathbb{K}}^{d}$ we have
\begin{equation}
|\cdot|_{q}\leq|\cdot|_{p}\leq d^{\frac{1}{p}-\frac{1}{q}}|\cdot
|_{q},\label{equ:lpnorms
\end{equation}
for all $1\leq p\leq q$. Since the dimension of $\mathcal{P}(^{m}\ell_{\infty
}^{n})$ is ${\binom{{m+n-1}}{{n-1}}}$, the result follows from $|P|_{2
\leq\Vert P\Vert$ by setting in \eqref{equ:lpnorms} $p=\frac{2m}{m+1}$, $q=2$
and $d={\binom{{m+n-1}}{{n-1}}}$. So $D_{m}={{\binom{{m+n-1}}{{n-1}}}}^{\frac{1}{2m}}$ and sinc
\[
\lim_{m\rightarrow\infty}{{\binom{{m+n-1}}{{n-1}}}}^{\frac{1}{2m}}=1
\]
the proof is done.
\end{proof}


\section{Comparative tables}

In this final section we organize the results of the previous sections in comparative tables below (for, respectively, polynomials and multilinear forms). We also add recent results for the constants of the polynomial Bohnenblust--Hille inequality and for the constants of the multilinear Bohnenblust--Hille inequality (see \cite{Nun2}).

\bigskip

\noindent \underline{Polynomials}:
\
\begin{tabular}
[t]{|c|c|c|}\hline
$\mathbb{K}$ & $\mathbb{R}$ & $\mathbb{C}$\\\hline
{\small {Optimal exponent}} & $2m/\left(  m+1\right)  $ & $2m/\left(
m+1\right)  $\\\hline
{\small Optimal e{xtra factor for} $r\in\lbrack1,\frac{2m}{m+1}]$} &
$n^{\left(  \frac{m}{r}-\frac{m+1}{2}\right)  }$ & $n^{\left(  \frac{m
{r}-\frac{m+1}{2}\right)  }$\\\hline
{\small {Hypercontractivity}} & \textsc{yes} & \textsc{yes}\\\hline
{\small {Optimality of the hypercontractivity}} & \textsc{yes} & ?\\\hline
{\small {Contractivity in} $\mathbb{K}^{n}$} & \textsc{no} & \textsc{yes
\\\hline
$H_{\infty,\mathbb{K}}$ & $\in\left[  1.50,2.83\right]  $ & $\leq\sqrt{2
$\\\hline
$\rho_{\mathbb{K}}$ & $\infty$ & $2$\\\hline
\end{tabular}
\ \ \
\]

\bigskip

\noindent \underline{Multilinear forms}:
\
\begin{tabular}
[t]{|c|c|c|}\hline
$\mathbb{K}$ & $\mathbb{R}$ & $\mathbb{C}$\\\hline
{\small {Optimal exponent}} & $2m/\left(  m+1\right)  $ & $2m/\left(
m+1\right)  $\\\hline
{\small {Optimal extra factor for} $r\in\lbrack1,\frac{2m}{m+1}]$} &
$n^{\left(  \frac{m}{r}-\frac{m+1}{2}\right)  }$ & $n^{\left(  \frac{m
{r}-\frac{m+1}{2}\right)  }$\\\hline
$\mu_{\mathbb{K}}$ & $2$ & $\leq2$\\\hline
$C_{\mathbb{K},m}$ & $<1.65\left(  n-1\right)  ^{0.53}+0.13$ & $<1.41\left(
n-1\right)  ^{0.31}-0.04$\\\hline
$C_{\mathbb{K},m}$ & $\geq2^{1-\frac{1}{m}}$ & $\geq1$\\\hline
$C_{\mathbb{K},2}$ & $\sqrt{2}$ & $\leq\frac{2}{\sqrt{\pi}}$\\\hline
\end{tabular}
\
\]

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