Some Phragmén-Lindelöf and harmonic majorization theorems for subharmonic functions (Q801440)

Let s be a subharmonic function in an unbounded domain \(\Omega \subset R^ n\) with boundary \(\partial \Omega \subset R^ n\) such that \(\lim_{M\to N,}\sup_{M\in \Omega}s(M)\leq 0,\) \(\forall N\in \partial \Omega\). A Phragmén-Lindelöf theorem asserts, under an additional PL- hypothesis, that \(s\leq 0\) in \(\Omega\). The present paper contains several results of this sort, where the novelty lies in the nature of the PL- hypothesis. This is briefly as follows. It is assumed that \(\Omega\) can be exhausted by a sequence of subdomains \(\Omega_ 1\subset \Omega_ 2\subset...\), subject to mild restrictions. Let \(\mu_{m,M}\) denote harmonic measure on \(\partial \Omega_ m\) relative to a point \(M\in \Omega_ m\); let H be positive and harmonic in \(\Omega_ 1\) and let be \(P\in \Omega_ 1\). Then it is shown that the sequence of numbers \[ \lambda_ m(s)=(H(P))^{-1}\int_{\partial \Omega_ m\cap \Omega}s d\mu_{m,P} \] converges to a limit \(\Lambda(s)\), \(-\infty <\Lambda (s)<+\infty\). (The authors prove a more general result.) Under mild assumptions on a real function \(\phi\), the PL-hypothesis \(\Lambda(\phi\circ s)\leq 0\), or \(<+\infty\), does imply \(s\leq 0\). There are several other results of considerable independent interest, including some about harmonic majorants. Examples and counterexamples for \(\Omega =R^ n\), a half-space or a strip are given. The techniques involve various notions such as R. S. Martin boundary points, methods of Perron- Wiener-Brelot in the Dirichlet problem and rely upon results of \textit{L. Naïm} [Ann. Inst. Fourier 7, 183-281 (1958; Zbl 0086.306)] and \textit{J.-M. Wu} [ibid. 28, No.4, 147-167 (1978; Zbl 0368.31006)].