On zero varieties of holomorphic functions in Hardy spaces

AbstractIn contrast to the famous Henkin–Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball Bn in Cn, n⩾2, it is proved in this article that for any nonnegative, increasing, convex function ϕ(t) defined on R, there exists g∈O(Bn) satisfying ∫Sϕ(Ng(ζ,1))dσ(ζ)<∞ such that there is no f∈Hp(Bn), 0<p<∞, with Z(f)=Z(g). Here Ng(ζ,1) denotes the integrated zero counting function associated with the slice function gζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces Hp(Bn), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner