Minimally fine limits at infinity for \(p\)-precise functions
If \(u\geq 0\) is a superharmonic function in the half-plane \(D= \mathbb{R}^{n-1}\times \{x_n>0\}\), then \(u(x)=ax_n+G\mu (x)+P\nu (x)\), where the constant \(a\geq 0\) and \(G\mu\) and \(P\nu\) are, respectively, Green and Poisson potentials of nonnegative measures \(\mu\) and \(\nu\). This representation implies that \(\lim u(x)/x_n=a\) when \(D\ni x\rightarrow \infty\) outside an exceptional set that is minimally thin at infinity in \(D\). Considering more refined characteristics of an exceptional set, one can make this statement more precise and, in particular, estimate the rate of convergence of \(u(x)/x_n\) to \(a\) [see, e.g., \textit{H. Aikawa} and \textit{M. Essén}, Potential theory -- selected topics (1996; Zbl 0865.31001)]. The present authors extend these results to locally \(p\)-precise functions in \(D\) satisfying \(\int _D | \nabla u| ^p x_n^{\gamma}\,dx<\infty\), \(0<p<\infty\), \(-1<\gamma <p-1\). Here a function \(u\in W^1_p(\Omega)\) is called \(p\)-precise in a bounded domain \(\Omega\) [\textit{W. Ziemer}, Mich. Math. J. 17, 117--128 (1970; Zbl 0183.39104)] if for each \(\varepsilon >0\), there exists an open set \(U\) such that \(u| _{\Omega \setminus U}\) is continuous and \(\psi _p(U)<\varepsilon\), where \((\psi _p(U))^{-1}\) is the \(p\)-dimensional extremal length of all closed connected sets joining \(U\) to infinity at \(\mathbb{R}^n\).
scientific article; zbMATH DE number 2092004