Abstract
We establish new characterizations of the Bloch space \(\mathcal {B}\) which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function \(f(z)=\sum _{n=0}^\infty \widehat{f}(n) z^n\) in the unit disc \(\mathbb {D}\), we define the fractional derivative \( D^{\mu }(f)(z)=\sum \limits _{n=0}^{\infty } \frac{\widehat{f}(n)}{\mu _{2n+1}} z^n \) induced by a radial weight \(\mu \), where \(\mu _{2n+1}=\int _0^1 r^{2n+1}\mu (r)\,dr\) are the odd moments of \(\mu \). Then, we consider the space \( \mathcal {B}^\mu \) of analytic functions f in \(\mathbb {D}\) such that \(\Vert f\Vert _{\mathcal {B}^\mu }=\sup _{z\in \mathbb {D}} \widehat{\mu }(z)|D^\mu (f)(z)|<\infty \), where \(\widehat{\mu }(z)=\int _{|z|}^1 \mu (s)\,ds\). We prove that \(\mathcal {B}^\mu \) is continously embedded in \(\mathcal {B}\) for any radial weight \(\mu \), and \(\mathcal {B}=\mathcal {B}^\mu \) if and only if \(\mu \in \mathcal {D}=\widehat{\mathcal {D}}\cap \check{\mathcal {D}}\). A radial weight \(\mu \in \widehat{\mathcal {D}}\) if \(\sup _{0\le r<1}\frac{\widehat{\mu }(r)}{\widehat{\mu }\left( \frac{1+r}{2}\right) }<\infty \) and a radial weight \(\mu \in \check{\mathcal {D}}\) if there exist \(K=K(\mu )>1\) such that \(\inf _{0\le r<1}\frac{\widehat{\mu }(r)}{\widehat{\mu }\left( 1-\frac{1-r}{K}\right) }>1.\)
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This research was supported in part by Ministerio de Ciencia e Innovación, Spain, project PID2022-136619NB-I00; La Junta de Andalucía, project FQM210.
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Funding for open access publishing: Universidad Málaga/CBUA. This research was supported in part by Ministerio de Ciencia e Innovación, Spain, project PID2022-136619NB-I00; La Junta de Andalucía, project FQM210.
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A. M. Moreno, J. A. Peláez and E. de la Rosa have equally contributed in each part of the paper.
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Moreno, Á.M., Peláez, J.Á. & de la Rosa, E. Fractional Derivative Description of the Bloch Space. Potential Anal (2024). https://doi.org/10.1007/s11118-023-10119-z
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DOI: https://doi.org/10.1007/s11118-023-10119-z