Bohr Inequalities on Noncommutative Polydomains

Abstract

The goal of this paper is to study the Bohr phenomenon in the setting of free holomorphic functions on noncommutative regular polydomains $$\mathbf{D_f^m}$$ , $$\mathbf{f}=(f_1,\ldots , f_k)$$ , generated by positive regular free holomorphic functions. These polydomains are noncommutative analogues of the scalar polydomains $$\begin{aligned} {{\mathcal {D}}}_{f_1} ({{\mathbb {C}}})\times \cdots \times {{\mathcal {D}}}_{f_k}({{\mathbb {C}}}), \end{aligned}$$ where each $${{\mathcal {D}}}_{f_i}({{\mathbb {C}}})\subset {{\mathbb {C}}}^{n_i}$$ is a certain Reinhardt domain generated by $$f_i$$ . We characterize the free holomorphic functions on $$\mathbf{D_f^m}$$ in terms of the universal model of the polydomain and extend several classical results from complex analysis to our noncommutative setting. It is shown that the free holomorphic functions admit multi-homogeneous and homogeneous expansions as power series in several variables. With respect to these expansions, we introduce the Bohr radii $$K_{mh}(\mathbf{D_f^m})$$ and $$K_{h}(\mathbf{D_f^m})$$ for the noncommutative Hardy space $$H^\infty (\mathbf{D_{f,\text { rad}}^m})$$ of all bounded free holomorphic functions on the radial part of $$\mathbf{D_f^m}$$ . Several well-known results concerning the Bohr radius associated with classes of bounded holomorphic functions are extended to our noncommutative multivariable setting.