Zero sets of holomorphic functions in the bidisc

In this work we characterize the zero sets of holomorphic functions f in the bidisc such that log jf j Lp D p Moreover we give a su cient condition on a analytic variety to be de ned by a function in A D Introduction In this paper we study some geometrical conditions on analytic varieties in the bidisc D fz C jz j jz j g to be de ned by an holomorphic func tion with some restriction on its growth In a strictly pseudo convex domain this kind of problems are better understood and for instance a complete char acterization of the zero sets of holomorphic functions in the Nevanlinna class see Khe Sko is known In the bidisc much less is known Nevertheless there are classes of functions whose zero sets had been characterized For instance the class of holomor phic functions such that log jf j L D see Cha and And In the second section we consider a variant of this problem namely functions such that log jf j L D We obtain a complete characterization of the zero sets of this class This problem is closely related to one considered by Beller in Bel in one variable where he studied the zero sequences of functions such that log jf j L D and in section we extend some results on zero sets due to Korenblum Kor where he characterizes the sequences of zeros of functions of slow growth in the disc that is holomorphic functions such that f explodes like a power of the distance to the border jf j C jzj nf First we discuss which is the natural de nition of A D and afterwards we give a su cient condition on the variety in order to be de ned by a function f A D Partially supported by the DGCYT grant PB C and grant GRQ from the Comissionat per Universitats i Recerca de la Generalitat de Catalunya Acknowledgments This work is part of my Ph D thesis and I want to express my sincere gratitude to my advisor Dr Joaquim Bruna for all his help and attention Zeros of functions with log jf j L D Statement of the results In this section we will give a complete characterization of the zero sets of holo morphic functions f H D such that log jf j L D Our main tool will be the Poincar e Lelong theorem Lel that shows that this problem is related to the problem of solving the equation i u with good estimates on u in terms of In order to state the theorem we need to introduce some notation Let be a closed positive current in the bidisc for any z D and xed let Dz be a small disc Dz D j zj j zj If is a closed positive current then it can expressed in coordinates as