Isomorphisms on Weighed Banach Spaces of Harmonic and Holomorphic Functions

For an arbitrary open subset U subset of R-d or U subset of C-d and a continuous function v : U ->]0,infinity[ we show that the space h(v0) (U) of weighed harmonic functions is almost isometric to a (closed) subspace of c(0), thus extending a theorem due to Bonet and Wolf for spaces of holomorphic functions H-v0 (U) on open sets U subset of C-d. Inspired by recent work of Boyd and Rueda, we characterize in terms of the extremal points of the dual of h(v0) (U) when h(v0) (U) is isometric to a subspace of c(0). Some geometric conditions on an open set U subset of C-d and convexity conditions on a weight v on U are given to ensure that neither H-v0 (U) nor h(v0) (U) are rotund.