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

TL;DR

This paper characterizes the dual of Hardy-type spaces on certain noncompact manifolds using BMO-like functions, establishes duality with finite atomic spaces, and explores solvability of the Poisson equation and properties of Bergman spaces.

Contribution

It introduces a dual space characterization for Hardy-type spaces on noncompact manifolds, linking it to BMO-like functions and atomic decompositions, with applications to operator boundedness.

Findings

Dual space Y^h(M) characterized by BMO-like functions.

Y^h(M) is dual to X^h(M) and X^k_fin(M).

Global solvability of the generalized Poisson equation proved.

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.