Uniqueness theorem and subharmonic test function

Abstract:

Suppose that a holomorphic function $f$ in a domain $D$ in $\mathbb {C}^n$ satisfies $|f|\leq e^M$ on $D$ (pointwise), where $M\not \equiv -\infty$ is subharmonic function on $D$ with Riesz measure $\nu _M$. Various methods are described for construction of wide classes of subharmonic test functions. These are subharmonic functions nonnegative and bounded on $D\setminus S_0$ for some compact set $S_0\subset D$ that have zero limit on the boundary of $D$. If the integral of such a test function over $D\setminus S_0$ with respect to the Riesz measure $\nu _M$ is finite and its integral with respect to the $(2n-2)$-dimensional Hausdorff measure over the zero set of $f$ is infinite, $f\equiv 0$ on $D$. Thus, any new test function yields a uniqueness theorem.