Hardy’s inequality for functions of several complex variables

Abstract

We obtain a generalization of Hardy’s inequality for functions in the Hardy space H1 (Bd), where Bd is the unit ball {z = (z1, …, zd) ∈ In particular, we construct a function φ on the set of d –dimensional multi-indices {n = (n1, …, nd) | ni ∈   {0}} and prove that if f(z) = Σ anzn is a function in H1 (Bd), then ≤ Moreover, our proof shows that this inequality is also valid for functions in Hardy space on the polydisk H1 (Bd).