RT Journal Article
T1 Fractional Volterra-type operator induced by radial weight acting on Hardy space
A1 Bellavita, Carlo
A1 Moreno, Álvaro Miguel
A1 Nikolaidis, Georgios
A1 Peláez-Márquez, José Ángel
K1 Hardy, Espacios de
K1 Volterra, Operadores de
AB Given  a radial doubling  weight $\mu$ on the unit disc $\mathbb{D}$ of the complex plane and its odd moments  $\mu_{2n+1}=\int_0^1 s^{2n+1}\mu(s)\, ds$, we consider the fractional derivative	$$	D^\mu(f)(z)=\sum_{n=0}^{\infty} \frac{\widehat{f}(n)}{\mu_{2n+1}}z^n, %\quad z\in \D,	$$ of  a function $ f(z)=\sum_{n=0}^{\infty}\widehat{f}(n)z^n$ analytic in $\mathbb{D}$. We  also  consider the fractional integral operator	$	I^\mu(f)(z)=\sum_{n=0}^{\infty} \mu_{2n+1}\widehat{f}(n)z^n, %\, z\in\D,	$ and the fractional  Volterra-type operator 	$$	V_{\mu,g}(f)(z)= I^\mu(f\cdot D^\mu(g))(z),\quad f\in\H(\D),%,\quad z\in \D.	$$ for any fixed $g\in\H(\D)$.We prove that $V_{\mu,g}$ is bounded (compact) on a Hardy space $H^p$, $0<p<\infty$, if and only  if $g$  belongs to $\BMOA$ ($\VMOA$).  Moreover, if $\int_0^1 \frac{\left(\int_r^1 \mu(s)\, ds\right)^p}{(1-r)^2}\,dr=+\infty$, we prove that  $V_{\mu,g}$ belongs to the Schatten class $S_p(H^2)$ if and only if $g=0$. On the  other hand,  if $\frac{\left(\int_r^1 \mu(s)\, ds\right)^p}{(1-r)^2}$ is a radial doubling weight it is proved that $V_{\mu,g} \in S_p(H^2)$ if and only if $g$ belongs to the  Besov space $B_p$.  En route, we obtain descriptions of $H^p$, $\BMOA$, $\VMOA$ and $B_p$ in terms of the fractional derivative $D^\mu$.
PB Springer Nature
YR 2026
FD 2026-01-20
LK https://hdl.handle.net/10630/45341
UL https://hdl.handle.net/10630/45341
LA eng
NO C. Bellavita,  A.M. Moreno, G. Nikolaidis, J.A. Peláez, Fractional Volterra-type operator induced by radial weight acting on Hardy space, Math. Z. {\bf 312} (2026), no.~2, Paper No. 49 https://doi.org/10.1007/s00209-025-03934-0
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 3 mar 2026