Bounded operators on the weighted spaces of holomorphic functions on the unit ball in $C^n$

TL;DR

This paper investigates boundedness of operators on weighted Besov spaces of holomorphic functions on the unit ball in complex n-space, using fractional derivatives and regular variation weights.

Contribution

It establishes conditions under which operators are bounded on these weighted Besov spaces, extending the understanding of operator behavior in complex analysis.

Findings

Operators are bounded on weighted Besov spaces with regular variation weights.

The paper characterizes boundedness conditions for fractional derivative operators.

Results apply to a broad class of weights in the space S.

Abstract

Assuming that $S$ is the space of functions of regular variation, $\omega\in S$, $0< p<\infty$, a function $f$ holomorphic in $B^n$ is said to be of Besov space $B_p(\omega)$ if $$\|f\|^p_{B_p(\omega )}=\int_{B^n} (1-|z|^2)^p|Df(z)|^p\frac{\omega(1-|z|)}{(1-|z|^2)^{n+1}}d\nu(z) <+\infty,$$ where $d\nu (z) $ is the volume measure on $B^n$ and $D $ stands for a fractional derivative of $f$. We consider operators on $B_p(\omega)$ and show, that they are bounded.