Isomorphisms on Weighed Banach Spaces of Harmonic and Holomorphic Functions

Abstract

For an arbitrary open subset U⊂ℝd or U⊆ℂd and a continuous function v:U→]0,∞[ we show that the space hv0(U) of weighed harmonic functions is almost isometric to a (closed) subspace of c0, thus extending a theorem due to Bonet and Wolf for spaces of holomorphic functions Hv0(U) on open sets U⊂ℂd. Inspired by recent work of Boyd and Rueda, we characterize in terms of the extremal points of the dual of hv0(U) when hv0(U) is isometric to a subspace of c0. Some geometric conditions on an open set U⊆ℂd and convexity conditions on a weight v on U are given to ensure that neither Hv0(U) nor hv0(U) are rotund.