Local and global majorization of subharmonic functions (Q791716)

Let s be subharmonic in a bounded domain in Euclidean space \(R^ n\) (\(n\geq 2)\). The question of whether the existence of a global harmonic majorant for s is equivalent to the existence of a harmonic majorant on a neighbourhood of each boundary point has received some attention in recent years. In particular, it follows from the work of \textit{P. M. Gauthier} and \textit{M. Goldstein} [J. Lond. Math. Soc., II. Ser. 16, 458- 466 (1977; Zbl 0379.31006)] that equivalence holds for Lipschitz domains. The present paper considers the same question for two special types of harmonic majorization, namely (i) majorization by a quasi-bounded harmonic function, as defined by \textit{M. Parreau} [Ann. Inst. Fourier 3, 103-197 (1952; Zbl 0047.320)], and (ii) majorization by the Perron- Wiener-Brelot solution of the generalized Dirichlet problem for the corresponding open set. Equivalence of local and global majorization is again established for Lipschitz domains, and an n-dimensional counter- example is provided to show that this result does not hold for arbitrary bounded domains.