Let $R$ be a smooth Riemannian manifold of finite volume, $\Delta$ its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of $R$ are found, and for biharmonic functions (those for which $\Delta \Delta u=0$) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.

Let $R$ be a Riemannian manifold without a biharmonic Green function defined on it and $\Omega$ a domain in $R$. A necessary and sufficient condition is given for the existence of a biharmonic Green function on $\Omega$.

Let $(X,\mathscr{H})$ and $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from $(X,\mathscr{H})$ to $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ is a continuous map from $X$ to $X^{\text{'}}$ which preserves the biharmonic structures of $X$ and $X^{\text{'}}$. In the present work we study this notion and characterize in some cases the biharmonic morphisms between $X$ and $X^{\text{'}}$ in terms of harmonic morphisms between the harmonic spaces associated with $(X,\mathscr{H})$ and $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ and the coupling kernels of them.