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 L^2_{loc}(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.

Harmonic Bergman spaces, the Poisson equation and the dual of Hardy-type spaces on certain noncompact manifolds / Mauceri, Giancarlo; Meda, Stefano; Vallarino, Maria. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - STAMPA. - 14:5(2015), pp. 1157-1188. [10.2422/2036-2145.201301_006]

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

VALLARINO, MARIA
2015

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 L^2_{loc}(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2650256
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