On von Neumann's inequality on the polydisc

Given a $d$-tuple $T$ of commuting contractions on Hilbert space and a polynomial $p$ in $d$-variables, we seek upper bounds for the norm of the operator $p(T)$. Results of von Neumann and And\^o show that if $d=1$ or $d=2$, the upper bound $\|p(T)\| \le \|p\|_\infty$, holds, where the supremum norm is taken over the polydisc $\mathbb{D}^d$. We show that for $d=3$, there exists a universal constant $C$ such that $\|p(T)\| \le C \|p\|_\infty$ for every homogeneous polynomial $p$. We also show that for general $d$ and arbitrary polynomials, the norm $\|p(T)\|$ is dominated by a certain Besov-type norm of $p$.