The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

We study the $L^p$-boundedness of the Berezin transform on the generalised Hartogs triangles which are defined by

$$ \begin{align*}H_k:=\{(z, w)\in\mathbb C^n\times\mathbb C: |z_1|^2+\cdots+|z_n|^2<|w|^{2k}<1\},\end{align*} $$

where $z=(z_1, \ldots , z_n)$ and $k\in \mathbb N$. We prove that the Berezin transform is bounded on $L^p(H_k)$ if and only if $p>nk+1$.

It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation

$$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$

${\it\mu}$

$|{\it\mu}|(\mathbb{D})<\infty$

$B_{p}$

$0<p\leq 1$

${\rm\Lambda}_{t}$

$t>1$

Let $\left( X,\,B,\,\mu \right)$ be a $\sigma $-finite measure space and let $H\,\subset \,{{L}^{2}}\left( X,\,\mu \right)$ be a separable reproducing kernel Hilbert space on $X$. We show that the multiplier algebra of $H$ has property $\left( {{A}_{1}}\left( 1 \right) \right)$.