Thin Sets at the Boundary

In his letter [18], Hayman raised the following questions: If u is negative subharmonic in the unit disk, then for almost all φ, u(z) has a limit as z tends to e nontangentially and outside a near-thin set ([17, Chapter 7]). There ought to be a global result, i.e. not in an angle, outside a suitable exceptional set. What should it be? It might be easier for Green potentials, where the limit is always zero. Of course the questions also exist in higher dimensions. The purpose of this paper is to give an answer to Hayman’s questions. We shall show that there are exceptional sets suitable for the description of (especially tangential) boundary behavior of superharmonic functions. As a result we shall prove that the above nontangential restriction can be relaxed (see Corollary 1.1). Let us first recall the notion of minimal thinness. This notion can be defined in a very general framework. Let D be a Martin space with Martin boundary ∂MD. A set E ⊂ D is called minimally thin at a minimal boundary point X if the reduced function over E of the Martin kernel at X is a potential. The Fatou–Naim–Doob theorem describes the boundary behavior of nonnegative superharmonic functions (or equivalently nonpositive subharmonic functions) with minimally thin sets.