Berezin regularity of domains in $\mathbb {C}^n$ and the essential norms of Toeplitz operators

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