Given a Banach space $X$, for $n\in \mathbb{N}$ and $p\in (1,\infty)$ we
investigate the smallest constant $\mathfrak P\!\in \!(0,\infty)$ for
which every $n$-tuple of functions $f_1,\ldots,f_n\!:\!\{-1,1\}^n\!\to\! X$
satisfies
\[\def\e{\varepsilon}\def\d{\delta}\int_{\{-1,1\}^n}\Big\|\sum_{j=1}^n
\partial_jf_j(\e)\Big\|^p\,d\mu(\varepsilon)\le
\mathfrak{P}^p\int_{\{-1,1\}^n}\int_{\{-1,1\}^n}\Big\|\sum_{j=1}^n
\d_j\varDelta f_j(\varepsilon)\Big\|^p\,d\mu(\varepsilon) \,d\mu(\delta),
\]
where $\mu$ is the uniform probability measure on the discrete
hypercube $\{-1,1\}^n$, and $\{\partial_j\}_{j=1}^n$ and
$\varDelta=\sum_{j=1}^n\partial_j$ are the hypercube partial
derivatives and the hypercube Laplacian, respectively. Denoting this
constant by $\mathfrak{P}_p^n(X)$, we show that
$$\mathfrak{P}_p^n(X)\le \sum_{k=1}^{n}\frac{1}{k}$$ for every Banach
space $(X,\|\cdot\|)$. This extends the classical Pisier inequality,
which corresponds to the special case $f_j=\varDelta^{-1}\partial_j f$
for some $f:\{-1,1\}^n\to X$. We show that $\sup_{n\in
\mathbb{N}}\mathfrak{P}_p^n(X)\infty$ if either the dual $X^*$ is a
$\mathrm{UMD}^+$ Banach space, or for some $\theta\in (0,1)$ we have
$X=[H,Y]_\theta$, where $H$ is a Hilbert space and $Y$ is an
arbitrary Banach space. It follows that $\sup_{n\in
\mathbb N}\mathfrak{P}_p^n(X)\infty$ if $X$ is a Banach lattice of
nontrivial type.
