Uniqueness theorems for subharmonic functions in unbounded domains

Gardiner, S. J.

doi:10.1090/S0002-9939-1987-0875377-7

Uniqueness theorems for subharmonic functions in unbounded domains
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Proc. Amer. Math. Soc. 99 (1987), 437-444 Request permission

Abstract:

A theorem of Carlson says that a holomorphic function of exponential growth in the half-plane cannot approach zero exponentially along the boundary unless it vanishes identically. This paper presents a generalization of this result for subharmonic functions in a Greenian domain $\Omega$, using the Martin boundary, minimal fine topology and PWB solution to the $h$-Dirichlet problem. Applications of the general theorem to specific choices of $\Omega$, such as the half-space and strip, are presented in later sections.

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