Unbounded harmonic functions on homogeneous manifolds of negative curvature (Q2773280)

Let the homogeneous manifold \(M\) of negative curvature be a semi-direct product \(N{\times}_sA\) of a nilpotent Lie group \(N\) and \(A={\mathbb R}^+\). Consider the second order differential operator \(L^{\gamma}=\sum_{j}(X^a_j)^2+X^a+a^2{\partial}^2_a+(1-\gamma)a{\partial}_a\), where \(\{X_j\}_{1\leq j\leq m}\) is a basis of the Lie algebra \(\underline{N}\) of \(N\) and \(X{\in}{\underline{N}}\), \(X^a=Ad(\exp(\log a)H)(X)\). A function \(F\) on \(M\) is called \(L^{\gamma}\)-harmonic if \(L^{\gamma}F=0\). This paper investigates unbounded harmonic functions and shows that if \(F\) has moderate growth, i.e., for all compact \(K{\subset}N\) and some \(r{\in}{\mathbb R}\), \(\int_K|F(x,a)|dx{\leq}C_Ke^{-r}\), then \(F\) has an asymptotic expansion as \(a{\rightarrow}0\) with coefficients in \(D'(N)\), the space of all distributions on \(N\), and \(F\) is uniquely determined by the two coefficients \(F_0(.)={\lim}_{a{\rightarrow}0^+}F(.,a)\) and \(G_0(.)={\lim}_{a{\rightarrow}0^+}a^{-\gamma}(F(.,a)-F_0(.))\) for \(0<{\gamma}{\leq}1\). In the case \({\gamma}=0\) a similar conclusion is also valid.NEWLINENEWLINENEWLINEUsing the asymptotic expansion of \(F\), the authors obtain certain sufficient conditions which assure that \(F\) can be written as a Poisson integral of some function \(f\) on \(N\). For example if \(F\) is \(L^{\gamma}\)-harmonic with \(0<{\gamma}{\leq}1\) and has moderate growth, and \(f=\|P^{\gamma}\|^{-1}_1F_0{\in}L^1((1+|x|)^{-Q-\gamma} dx)\), then \(F=P^{\gamma}(f)\) if and only if \(G_0(x)=\int_N(f(xy)-f(x)){\rho}_{\gamma}(y)^{-Q-\gamma} dy\), where \(P^{\gamma}\) is the Poisson kernel.