Volume growth, Green's functions, and parabolicity of ends (Q1974904)

Let \(M\) be a complete, noncompact Riemannian \(n\)-manifold without boundary, \(V(t)\) be the volume of the geodesic ball \(B(o,t)\) of radius \(t\) centered at the fixed point \(o\in M\). If \(C\subset M\) is a compact subset, then an unbounded component of \(M\backslash C\) is called an end with respect to \(C\). The author considers the generalized \(p\)-harmonic functions satisfying the equation div\(A(\nabla u)=0\) in the Sobolev space \(W^{1,p}\), where \(A\) is a differential operator. He introduces the notation of large (small) end \(E\) when \(\int_{R_{0}}^{\infty }[ \frac{t}{V_{E}(t)}] ^{1/(p-1)} dt<\infty\) \((=\infty)\), where \(V_{E}(t)=|E\cap B(o,t)|\), \(R_{0}\) is a sufficiently large number. It is proved that every \(p\)-small end is \(p\)-parabolic, i.e. it has zero \(p\)-capacity.