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Dimension-free H∞-calculus of angle <π/2 for UMD-valued Ornstein--Uhlenbeck operators

  • Jan van Neerven
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Abstract

Let 1<p<∞, let X be a UMD Banach space, let 1<p<∞, and let Ld be the generator of the Ornstein--Uhlenbeck semigroup (Pd(t))t≥ 0 on Lp(ℝd,γd;X). We prove that the operators −Ld are R-sectorial with a common angle strictly smaller than π/2 and with bounds independent of d. Combining this with the Hieber--Prüss transference theorem and the Kalton--Weis angle comparison, we deduce that the operators −Ld admit bounded H∞-calculi of a common angle strictly smaller than π/2, again with dimension-free bounds. The corresponding Walsh R-analyticity estimates are proved first and transferred to the Ornstein--Uhlenbeck setting by a central limit argument.

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Cite

BibTeXRIS
BibTeX
@misc{neerven2026dimensionfree,
  title = {{Dimension-free $H^\infty$-calculus of angle $<π/2$ for UMD-valued Ornstein--Uhlenbeck operators}},
  author = {Jan van Neerven},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04974},
  eprint = {2609.04974},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04974},
}
RIS
TY  - UNPB
TI  - Dimension-free H^∞-calculus of angle <π/2 for UMD-valued Ornstein--Uhlenbeck operators
AU  - Jan van Neerven
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04974
UR  - https://arxiv.org/abs/2609.04974
AB  - Let 1<p<∞, let X be a UMD Banach space, let 1<p<∞, and let L_d be the generator of the Ornstein--Uhlenbeck semigroup (P_d(t))_t≥ 0 on L^p(ℝ^d,γ_d;X). We prove that the operators −L_d are R-sectorial with a common angle strictly smaller than π/2 and with bounds independent of d. Combining this with the Hieber--Prüss transference theorem and the Kalton--Weis angle comparison, we deduce that the operators −L_d admit bounded H^∞-calculi of a common angle strictly smaller than π/2, again with dimension-free bounds. The corresponding Walsh R-analyticity estimates are proved first and transferred to the Ornstein--Uhlenbeck setting by a central limit argument.
ER  -