On the Distribution of Zero Sets of Holomorphic Functions

Аннотация

Let M be a subharmonic function with Riesz measure ν_M in a domain D in the n-dimensional complex Euclidean space ℂ_n, and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ⊂ D, and satisfies |f| ⩽ expM on D. Then restrictions on the growth of ν_M near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.