Small weighted Bergman spaces

This paper is based on the course \lq\lq Weighted Hardy-Bergman spaces\rq\rq\, I delivered in the Summer School \lq\lq Complex and Harmonic Analysis and Related Topics\rq\rq at the Mekrijärvi research station of University of Eastern Finland, June $2014$. The main purpose of this survey is to present recent progress on the theory of Bergman spaces $A^p_ω$, induced by radial weights $ω$ satisfying the doubling property $\int_r^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$.

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