Fractional Volterra-type operator induced by radial weight acting on Hardy space

TL;DR

This paper studies a fractional Volterra-type operator on Hardy spaces induced by radial weights, characterizing its boundedness, compactness, and Schatten class membership in terms of function spaces like BMOA, VMOA, and Besov spaces.

Contribution

It provides new characterizations of the fractional Volterra operator's properties on Hardy spaces using BMOA, VMOA, and Besov spaces, extending previous operator theory results.

Findings

V_{ extmu,g} is bounded on H^p iff g belongs to BMOA.

V_{ extmu,g} is compact on H^p iff g belongs to VMOA.

V_{ extmu,g} belongs to Schatten class S_p(H^2) iff g=0 or g belongs to B_p depending on weight conditions.

Abstract

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 $$ D^\mu(f)(z)=\sum_{n=0}^{\infty} \frac{\widehat{f}(n)}{\mu_{2n+1}}z^n, $$ 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 $I^\mu(f)(z)=\sum_{n=0}^{\infty} \mu_{2n+1}\widehat{f}(n)z^n$, and the fractional Volterra-type operator $$ V_{\mu,g}(f)(z)= I^\mu(f\cdot D^\mu(g))(z),\quad f\in\mathcal{H}(\mathbb{D}), $$ for any fixed $g\in\mathcal{H}(\mathbb{D})$. 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 $\text{BMOA}$ ($\text{VMOA}$). 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 $H^p$, $\text{BMOA}$, $\text{VMOA}$ and $B_p$ in terms of the fractional derivative $D^\mu$.