The Hilbert matrix operator into the Bergman space

@misc{norrbo2026hilbert,
  title = {{The Hilbert matrix operator into the Bergman space}},
  author = {David Norrbo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00252},
  eprint = {2610.00252},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00252},
}

TY  - UNPB
TI  - The Hilbert matrix operator into the Bergman space
AU  - David Norrbo
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00252
UR  - https://arxiv.org/abs/2610.00252
AB  - The Hilbert matrix operator H is bounded on the standard weighted Bergman spaces A^p_α, p>α+2>1. We examine the limiting case when α=p−2, and for p≤ 2, we characterize the logarithmic modification of the standard weight needed for the domain A^p_v_γ,p and codomain A^p_v_β,p to ensure boundedness. In particular, we prove that H:A^2_log^γ → A^2_log^β is bounded if and only if γ−β≥ 2 and γ>1, solving an open problem posed by B. Karapetrović and M. Jevtić. We also settle the situation for p>2 in all cases, except when the logarithmic exponents γ and β satisfy γ−β=p. Moreover, we completely solve the same problem when the domain and codomain are certain weighted reflexive ℓ^p spaces. All results are extended to the generalized Hilbert operator H_g.
ER  -