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The bounds ℏp,X≲(βp,X)2 and βp,X≲(ℏp,X)2 are sharp

  • Emiel Lorist
  • Jan van Neerven
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Abstract

It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space X and 1<p<∞, the UMDp property for X is equivalent to boundedness of the Hilbert transform on Lp(R;X), and that the UMD constant βp,X and the Hilbert transform constant ℏp,X are related by the quadratic bounds ℏp,X≲(βp,X)2, βp,X≲(ℏp,X)2. In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit 2n-dimensional Banach spaces for which the Hilbert transform constant grows like n and the UMD constant like √n, and a second family with the reverse behaviour.

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Cite

BibTeXRIS
BibTeX
@misc{lorist2026bounds,
  title = {{The bounds $\hbar_p,X\lesssim(β_p,X)^2$ and $β_p,X\lesssim(\hbar_p,X)^2$ are sharp}},
  author = {Emiel Lorist and Jan van Neerven},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10444},
  eprint = {2609.10444},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10444},
}
RIS
TY  - UNPB
TI  - The bounds ℏ_p,X≲(β_p,X)^2 and β_p,X≲(ℏ_p,X)^2 are sharp
AU  - Emiel Lorist
AU  - Jan van Neerven
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10444
UR  - https://arxiv.org/abs/2609.10444
AB  - It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space X and 1<p<∞, the UMD_p property for X is equivalent to boundedness of the Hilbert transform on L^p(R;X), and that the UMD constant β_p,X and the Hilbert transform constant ℏ_p,X are related by the quadratic bounds ℏ_p,X≲(β_p,X)^2, β_p,X≲(ℏ_p,X)^2. In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit 2^n-dimensional Banach spaces for which the Hilbert transform constant grows like n and the UMD constant like √n, and a second family with the reverse behaviour.
ER  -