Adem Ersin Üreyen

We study the Bloch and the little Bloch spaces of harmonic functions on the real hyperbolic ball. We show that the Bergman projections from $$L^\infty ({\mathbb {B}})$$L ∞ ( B ) to $${\mathcal {B}}$$B , and from $$C_0({\mathbb {B}})$$C 0 ( B ) to $${\mathcal {B}}_0$$B 0 are onto. We verify that the dual space of the hyperbolic harmonic Bergman space $${\mathcal {B}}^1_\alpha $$B α 1 is $${\mathcal {B}}$$B and its predual is $${\mathcal {B}}_0$$B 0 . Finally, we obtain atomic decompositions of Bloch functions as series of Bergman reproducing kernels.

We study the Bloch and the little Bloch spaces of harmonic functions on the real hyperbolic ball. We show that the Bergman projections from $L^\infty(\mathbb B)$ to $\mathcal B$, and from $C_0(\mathbb B)$ to $\mathcal B_0$ are onto. We verify that the dual space of the hyperbolic harmonic Bergman space $\mathcal B^1_α$ is $\mathcal B$ and its predual is $\mathcal B_0$. Finally, we obtain an atomic decomposition of Bloch functions as a series of Bergman reproducing kernels.

For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.