Abstract
In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball B <sub>n</sub> in ℂ <sup>n</sup> , n≥2, it is proved in this article that for any nonnegative, increasing, convex function (t) defined on ℝ, there exists g∈ O(B <sub>n</sub> ) satisfying ∫ <sub>S</sub> (N <sub>g</sub> (ζ,1)) dσ(ζ)<∞ such that there is no f∈H <sup>p</sup> (B <sub>n</sub> ), 0<p<∞, with Z(f)=Z(g). Here N <sub>g</sub> (ζ,1) denotes the integrated zero counting function associated with the slice function g <sub>ζ</sub> . This means that the zero sets of holomorphic functions belonging to the Hardy spaces H <sup>p</sup> (B <sub>n</sub> ), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner. © 2004 Elsevier Inc. All rights reserved.