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Littlewood-Paley theory for orthogonal expansions associated with root systems

  • Lukas Langen
  • Margit Rösler
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Abstract

We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley g-functions in the spirit of Stein's work for compact Lie groups and prove their Lp-boundedness for 1<p≤ 2 and for some of them also for 1<p<∞. As an application, we define associated Riesz transforms and imaginary powers and prove their Lp-continuity for 1<p<∞. Passing to the average with respect to the action of the associated reflection group, we obtain Lp-boundedness of g-functions for the symmetric Poisson semigroup for all 1<p<∞. In particular, our framework covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases.

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Cite

BibTeXRIS
BibTeX
@misc{langen2026littlewoodpaley,
  title = {{Littlewood-Paley theory for orthogonal expansions associated with root systems}},
  author = {Lukas Langen and Margit Rösler},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12722},
  eprint = {2609.12722},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12722},
}
RIS
TY  - UNPB
TI  - Littlewood-Paley theory for orthogonal expansions associated with root systems
AU  - Lukas Langen
AU  - Margit Rösler
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.12722
UR  - https://arxiv.org/abs/2609.12722
AB  - We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley g-functions in the spirit of Stein's work for compact Lie groups and prove their L^p-boundedness for 1<p≤ 2 and for some of them also for 1<p<∞. As an application, we define associated Riesz transforms and imaginary powers and prove their L^p-continuity for 1<p<∞. Passing to the average with respect to the action of the associated reflection group, we obtain L^p-boundedness of g-functions for the symmetric Poisson semigroup for all 1<p<∞. In particular, our framework covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases.
ER  -