Riesz-Martin representation for positive super-polyharmonic functions in a Riemannian manifold
It is known that if \(v\geq 0\) is a superharmonic function on a bounded domain \(\Omega\) in \(\mathbb{R}^n\), \(n\geq 2\), then \[ v(x)= \int_\Omega G(x,y) d\mu(y)+ \int_{\Delta_1} K(x,y) d\nu(y), \] \(x\in\Omega\), where \(G(x,y)\) is the Green kernel, \(K(x,y)\) is the Martin kernel, \(\mu\geq 0\) is a Radon measure on \(\Omega\) and \(\nu\geq 0\) is a Radon measure with support in the minimal boundary \(\Delta_1\) (which is contained in the Martin boundary \(\Delta\)). Thus, a superharmonic function \(v\geq 0\) on \(\Omega\) can be represented by two nonnegative Radon measures \(\mu\) on \(\Omega\) and \(\nu\) on Martin boundary, when the two kernels \(G(x,y)\) and \(K(x,y)\) are suitably fixed on \(\Omega\). This Riesz-Martin representation for \(v\) has been variously extended to domains \(\Omega\) in Riemannian manifolds and in locally compact spaces (in the axiomatic potential theory). This note shows that for a suitable domain \(\Omega\) in a Riemannian manifold, if \(v\) is a locally (volume-element) \(dx\)-integrable function such that \((-\Delta)^iv\geq 0\) on \(\Omega\) for \(0\leq i\leq m\), then \(v\) can be represented in the above sense (with respect to the Green kernel and suitably modified Martin kernels on \(\Omega\)) by a Radon measure \(\mu\geq 0\) on \(\Omega\) and \(m\) Radon measures \(\nu_i\geq 0\) on \(\Lambda_1\).
scientific article; zbMATH DE number 5164260
scientific article