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This work is an introduction to anisotropic spaces of holomorphic functions, which have $\omega$-weight and are generalizations of Bloch spaces on a unit ball. We describe the holomorphic Bloch space in terms of the corresponding $L_{\omega }^{\infty }$ space. We establish a description of ${\left(A^p\left(\omega \right)\right)}^{*}$ via the Bloch classes for all $0<p\le 1$.

The $\omega$-weighted Besov spaces of holomorphic functions on the unit ball $B^n$ in $C^n$ are introduced as follows. Given a function $\omega$ of regular variation and $0<p<\infty$, a function $f$ holomorphic in $B^n$ is said to belong to the Besov space $B_p\left(\omega \right)$ if $${\parallel f\parallel }_{B_p\left(\omega \right)}^p={\int }_{B^n}{(1-|z|}^2{)}^p{\left|Df\left(z\right)\right|}^p\frac{\omega (1-|z\left|\right)}{{(1-|z|}^2{)}^{n+1}}d\nu \left(z\right)<+\infty ,$$ where $d\nu \left(z\right)$ is the volume measure on $B^n$ and $D$ stands for the fractional derivative of $f$. The holomorphic Besov space is described in the terms of the corresponding $L_p\left(\omega \right)$ space. Some projection theorems and theorems on existence of the inversions of these projections are proved. Also,...