On Wiener Conditions for minimally thin and rarefied Sets
Matts R. Essén · Birkhäuser Basel eBooks · 1988
Let D = {x ∈ ℝ p : x 1 > 0}, where x = (x 1,…, x p ), p ≥ 2 and ∂D is the euclidean boundary of D. If u is subharmonic in D and y ∈ ∂D, we define u(y) = lim sup u(x), x → y, x ∈ D. If u is non-positive on ∂D and sup D u(x)/x 1 < ∞, it is known that $$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {u\left( x \right)/{x_1} \to \alpha ,}&{x \to \infty ,}&{x \in D\backslash E,} \end{array}} \\ {\begin{array}{*{20}{c}} {\left( {u\left( x \right) - \alpha {x_1}} \right)/\left| x \right| \to 0,}&{x \to \infty ,}&{x \in D\backslash F,} \end{array}} \end{array} $$ where the exceptional set E is minimally thin at infinity in D (cf. [5]) and the exceptional set F is rarefied at infinity in D (cf. [3]).