PreprintCC BY 4.0arXiv
The Differential Hilbert Operator Between Weighted Bergman Spaces
Abstract
In this paper, a complete characterization of the boundedness, compactness, norm and essential norm of the differential Hilbert operator ℋ2:A2α→ A2β is obtained. More precisely, ℋ2:A2α→ A2β is bounded if and only if −1<α<0 and β≥α+2 and it is compact if and only if −1<α<0 and β>α+2. The norm and essential norm of $ \|\mathcal H_2\|_{A^2_α\to A^2_β}$ are also investigated. In particular, when β=α+2, |ℋ2|A2α→ A2α+2 =|ℋ2|e,A2α→ A2α+2 =(π√((α+2)(α+3)))/(sin(π(α+2)/2)), −1<α<0. When β>α+2, |ℋ2|e,A2α→ A2β=0. Furthermore, for every 1≤ p<∞, the operator ℋ2 belongs to the Schatten class 𝒮p if and only if it is compact.
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BibTeX
@misc{li2026differential,
title = {{The Differential Hilbert Operator Between Weighted Bergman Spaces}},
author = {Songxiao Li and Mengmeng Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05672},
eprint = {2610.05672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05672},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Differential Hilbert Operator Between Weighted Bergman Spaces
AU - Songxiao Li
AU - Mengmeng Zhou
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05672
UR - https://arxiv.org/abs/2610.05672
AB - In this paper, a complete characterization of the boundedness, compactness, norm and essential norm of the differential Hilbert operator ℋ_2:A^2_α→ A^2_β is obtained. More precisely, ℋ_2:A^2_α→ A^2_β is bounded if and only if −1<α<0 and β≥α+2 and it is compact if and only if −1<α<0 and β>α+2. The norm and essential norm of $ \|\mathcal H_2\|_{A^2_α\to A^2_β}$ are also investigated. In particular, when β=α+2, |ℋ_2|_A^2_α→ A^2_α+2 =|ℋ_2|_e,A^2_α→ A^2_α+2 =(π√((α+2)(α+3)))/(sin(π(α+2)/2)), −1<α<0. When β>α+2, |ℋ_2|_e,A^2_α→ A^2_β=0. Furthermore, for every 1≤ p<∞, the operator ℋ_2 belongs to the Schatten class 𝒮_p if and only if it is compact.
ER -