Analytic Approximations and Dirichlet Problems on Bordered Riemann Surfaces
Stephen J. Scheinberg · Annals of Mathematics · 1975
Theorem 2 is proved from Theorem 1 in this fashion. If f is bounded, the classical Perron method as described in [1] gives a solution to the Dirichlet problem with boundary data f. If f is not bounded, then B is not compact; so M is not compact. Choose an analytic function F so that I f F I 0 is given, choose F1 so that I F1 -(1 + 1/7)1 1/p7; so I eFlI = eReF, > ells > 1/7. Now choose F. so that I elf F0 I < 1, and it follows that I fe-FFo I I e-Fl I < I. Theorem 1 will be a consequence of the preceding paragraph plus the next assertion.