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Essential norms of fractional Volterra-type operators
Abstract
Given a radial doubling weight μ on the unit disk 𝔻 and f(z)=∑n=0∞ f̂(n)zn∈ℋ(𝔻), the fractional derivative of f induced by μ is Dμ(f)(z) = ∑n=0∞ f̂(n)/(μ2n+1)zn, where μ2n+1=∫01 r2n+1μ(r) dr are the odd moments of μ. The fractional integral of f induced by μ is Iμ(f)(z)=∑n=0∞ μ2n+1f̂(n)zn. We study the fractional Volterra-type operator Vμ,g(f)(z)=Iμ(f· Dμ(g))(z), f∈ℋ(𝔻), with symbol g∈ℋ(𝔻), introduced by Bellavita, Moreno, Nikolaidis and Peláez. First, we estimate the essential norm of Vμ,g acting on Hp and Apα. Later, we study the same questions for Vμ,g:Hp→ Hq and Vμ,g:Apα→ Aqα, characterizing their boundedness and compactness and estimating their essential norms for all 0<p,q<∞.
Related
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BibTeX
@misc{moreno2026essential,
title = {{Essential norms of fractional Volterra-type operators}},
author = {Álvaro Miguel Moreno},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37548},
eprint = {2610.37548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37548},
}
RIS
TY - UNPB
TI - Essential norms of fractional Volterra-type operators
AU - Álvaro Miguel Moreno
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37548
UR - https://arxiv.org/abs/2610.37548
AB - Given a radial doubling weight μ on the unit disk 𝔻 and f(z)=∑_n=0^∞ f̂(n)z^n∈ℋ(𝔻), the fractional derivative of f induced by μ is D^μ(f)(z) = ∑_n=0^∞ f̂(n)/(μ_2n+1)z^n, where μ_2n+1=∫_0^1 r^2n+1μ(r) dr are the odd moments of μ. The fractional integral of f induced by μ is I^μ(f)(z)=∑_n=0^∞ μ_2n+1f̂(n)z^n. We study the fractional Volterra-type operator V_μ,g(f)(z)=I^μ(f· D^μ(g))(z), f∈ℋ(𝔻), with symbol g∈ℋ(𝔻), introduced by Bellavita, Moreno, Nikolaidis and Peláez. First, we estimate the essential norm of V_μ,g acting on H^p and A^p_α. Later, we study the same questions for V_μ,g:H^p→ H^q and V_μ,g:A^p_α→ A^q_α, characterizing their boundedness and compactness and estimating their essential norms for all 0<p,q<∞.
ER -