Semilinear perturbations of harmonic spaces and Martin-Orlicz capacities: an approach to the trace of moderate $U$-functions

Abstract

Let $(X,\mathcal H)$ be a harmonic space in the sense of H. Bauer [7] which has a Green function $G_X$. It is known [27] that to every reference measure $r$ there corresponds a suitable integral representation of functions in ${{\mathcal H}^+(X)}\cap L^1(X,r)$. Let $Y$ be the minimal Martin boundary, $P$ the Martin kernel, and denote by ${\mathcal M(Y)}$ the set of all signed Borel measures on $Y$ with bounded variation. In this paper we consider the perturbed (semilinear) structure $(X,\mathcal U)$ obtained from $(X,\mathcal H)$ by means of $({\gamma},\Psi)$ where ${\gamma}$ is a local Kato measure on $X$ and $\Psi$ belongs to a class of real-valued functions on $X\times{\mathbb R}$ containing $(x,t)\mapsto t|t|^{{\alpha}-1}$ for any ${\alpha}>1$. We show that for every function $u\in{\mathcal U_r(X)}:=\{u\in{\mathcal U(X)}:|u|\leq h\;\mbox{for some}\;h\in {{\mathcal H}^+(X)}\cap L^1(X,r)\}$ there exists a unique signed measure $\nu\in{\mathcal M(Y)}$ such that $$ u+\int_X G_X(\cdot,{\zeta})\Psi({\zeta},u({\zeta}))\,d{\gamma}({\zeta})=\int_Y P(\cdot,y)\,d\nu(y). $$ Conversely, we prove that this integral equation admits a solution $u\in{\mathcal U_r(X)}$ whenever $\nu$ does not charge compact sets $K\subset Y$ of zero Martin-Orlicz capacity, that is, the integral $$ \int_X\int_X G_X(x,{\zeta}) \Psi({\zeta},\int_Y P({\zeta},y)\,d\mu(y))\,d{\gamma}({\zeta})\,d r(x) $$ is equal to $0$ or $\infty$ for every $\mu\in{\mathcal M^+(Y)}$ such that $\mu(Y{\backslash} K)=0$. In Section 6, we use our approach to investigate the trace of moderate solutions to some semilinear equations.