Harmonic Bergman spaces, the Poisson equation and the dual of Hardy-type spaces on certain noncompact manifolds

2013/05/06 by G. Mauceri, Mauceri, G., S. Meda +3
Mathematics · #30H10 #42B20 #42B35 #58C99 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:30H10 #msc:42B20 #msc:42B35 #msc:58C99

paper · pdf · doi:10.48550/arxiv.1305.1127

31 pages

arxiv created 2013/05/06 · arxiv updated 2013/05/07

Abstract

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We realize the dual space Y^h(M) of the Hardy-type space X^h(M), introduced in a previous paper of the authors, as the class of all locally square integrable functions satisfying suitable BMO-like conditions, where the role of the constants is played by the space of global k-quasi-harmonic functions. Furthermore we prove that Y^h(M) is also the dual of the space X^k_fin(M) of finite linear combination of X^k-atoms. As a consequence, if Z is a Banach space and T is a Z-valued linear operator defined on X^k_fin(M), then T extends to a bounded operator from X^k(M) to Z if and only if it is uniformly bounded on X^k-atoms. To obtain these results we prove the global solvability of the generalized Poisson equation L^ku=f with f in L²_loc(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.

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