Bulletin of the
Korean Mathematical Society BKMS

On the setting of the half-space of the Euclidean $n$-space, we consider harmonic Bergman spaces and we also study properties of the reproducing kernel. Using covering lemma, we find some equivalent quantities. We prove that if $\displaystyle{ \lim_{ i \to \infty} \frac{ \mu \big(K_{r}(z_{i}) \big)}{ V \big( K_{r}(z_{i}) \big)}=0}$ then the inclusion function $\displaystyle{ I : b^{p} \rightarrow L^{p}(H_{n} , d\mu)}$ is a compact operator. Moreover, we show that if $f$ is a nonnegative continuous function in $L^{ \infty}$ and $\displaystyle{ \lim_{z \to \infty} f(z) =0}$, then $T_{f}$ is compact if and only if $f \in C_{0}(H_{n})$.

Keywords: Bergman spaces, reproducing kernels, Toeplitz operators, idempotent operators

MSC numbers: Primary 47B35, 47B38; Secondary 32A36