boundary growth theorems for superharmonic functions (Q1810330)

The paper examines the boundary behaviour of superharmonic functions \(u(X',x)\) on the half-space \({\mathbf{R}}^{n-1}\times (0,\infty)\) in terms of the behaviour along the lines normal to the boundary. In [Proc. Amer. Math. Soc. 124, No. 12, 3721-3727 (1996; Zbl 0868.31006)] the author has proved that if the set \[ E=\{X':\limsup_{x\to 0+}u(X',x)=+\infty\} \] is metrically fine dense (i.e., \(E\cap V\) has positive outer Lebesgue measure for every non-empty finely open set \(V\subset {\mathbf{R}}^{n-1}\)), then the set \[ \{X':\liminf_{x\to 0+}u(X',x)>-\infty\} \] is of first fine Baire category in \({\mathbf{R}}^{n-1}\). In the present paper, some refined versions of that result are obtained by considering the sets \[ E=\{X':\limsup_{x\to 0+}x^{n-1-\alpha}u(X',x)=+\infty\}. . \]