New properties of the multivariable $H^\infty$ functional calculus of sectorial operators

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 $\theta_k<\frac{\pi}{2}$.