$${\cal N}\left( {p,q,s} \right)$$-Type Spaces in the Unit Ball of ℂn. IV: Atomic Decomposition, Gleason’s Problem and Distance Problems

The purpose of this paper is to study several different properties of the holomorphic $${\cal N}\left( {p,q,s} \right)$$ -type functions in the unit ball $$\mathbb{B}$$ of ℂn via Carleson measure techniques. More precisely, we establish an atomic decomposition on such spaces and then we solve the Gleason’s problem on them. We also study the distance problems between Bergman-type spaces $${A^{ - {q \over p}}}\left(\mathbb{R} \right)$$ and $${\cal N}\left( {p,q,s} \right)$$ -type spaces. These results yield some new characterizations of the holomorphic F(p, q, t)-functions.