  This paper is devoted to the multivariable $H^\infty$ functional calculus
associated with a finite commuting family of sectorial operators on Banach
space. First we prove that if $(A_1,\ldots, A_d)$ is such a family, if $A_k$ is
$R$-sectorial of $R$-type $\omega_k\in(0,\pi)$, $k=1,\ldots,d$, and if
$(A_1,\ldots, A_d)$ admits a bounded $H^\infty(\Sigma_{\theta_1}\times
\cdots\times\Sigma_{\theta_d})$ joint functional calculus for some $\theta_k\in
(\omega_k,\pi)$, then it admits a bounded $H^\infty(\Sigma_{\theta_1}\times
\cdots\times\Sigma_{\theta_d})$ joint functional calculus for all $\theta_k\in
(\omega_k,\pi)$, $k=1,\ldots,d$. Second we introduce square functions adapted
to the multivariable case and extend to this setting some of the well-known
one-variable results relating the boundedness of $H^\infty$ functional calculus
to square function estimates. Third, on $K$-convex reflexive spaces, we
establish sharp dilation properties for $d$-tuples $(A_1,\ldots, A_d)$ which
admit a bounded $H^\infty(\Sigma_{\theta_1}\times
\cdots\times\Sigma_{\theta_d})$ joint functional calculus for some
$\theta_k<\frac{\pi}{2}$.
