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For the open unit disc $\mathbb{D}$ in the complex plane, it is well knownthat if $\phi \in C(\overline{\mathbb{D}})$ then its Berezin transform$\widetilde{\phi}$ also belongs to $C(\overline{\mathbb{D}})$. We say that$\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of somepseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometryplays an important role. We also establish a relationship between the essentialnorm of an operator in a natural Toeplitz subalgebra and its Berezin transform.

Berezin regularity of domains in C^n and the essential norms of Toeplitz operators