It is shown that the classical Bloch space and the Bergman space \(A^p_\eta \) , induced by a radial doubling weight \(\eta \) , can be characterized by using the fractional derivative \(\begin{aligned} f\mapsto R_{\nu ,\omega }(f)(z)=\sum _{k=0}^\infty \frac{\nu _{2k+1}}{\omega _{2k+1}}\widehat{f}(k)z^k,\quad z\in \mathbb {D}, \end{aligned}\) induced by two radial weights admitting certain doubling conditions. Here \(\begin{aligned} \omega _{2k+1}=\int _0^1r^{2k+1}\omega (r)\,dr \end{aligned}\) are the odd moments of \(\omega \) , and \(\widehat{f}(k)\) stands for the Maclaurin coefficient of the analytic function f in the unit disc \(\mathbb {D}\) . The main findings of this paper generalize and complete in part recent results by Moreno, Peláez and de la Rosa [Fractional derivative description of the Bloch space, Potential Anal. 61 (2024), no. 3, 555–571], and Peláez and de la Rosa [Littlewood-Paley inequalities for fractional derivative on Bergman spaces, Ann. Fenn. Math. 47 (2022), no. 2, 1109–1130]. The arguments employed here rely on integral representations via Bergman reproducing kernels of the operator \(R_{\nu ,\omega }\) and hence differ from those used in the said papers.
