Group-free radial Kunze-Stein phenomenon on affine buildings

Abstract

We give a geometric proof of the radial Kunze--Stein phenomenon on the good vertices of a thick, locally finite, regular affine building of irreducible type. The proof uses a positive spherical function, its joint eigenvalues for sphere averages, and the growth of vector-distance spheres. Conjugation by this function and complex interpolation yield a weighted ℓ² estimate and the boundedness of radial operators with ℓ^p kernels for 1≤ p<2. We also present proofs by the averaging operators norms, by the Schur test and by Helgason transform. A further interpolation gives weighted mixed-exponent estimates. No transitivity assumption on the automorphism group is required; in particular the argument applies to buildings of type Ã₂ without a Bruhat--Tits realization.

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BibTeX RIS

BibTeX

@misc{pawlak2026groupfree,
  title = {{Group-free radial Kunze-Stein phenomenon on affine buildings}},
  author = {Łukasz Pawlak},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07889},
  eprint = {2611.07889},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07889},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Group-free radial Kunze-Stein phenomenon on affine buildings
AU  - Łukasz Pawlak
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07889
UR  - https://arxiv.org/abs/2611.07889
AB  - We give a geometric proof of the radial Kunze--Stein phenomenon on the good vertices of a thick, locally finite, regular affine building of irreducible type. The proof uses a positive spherical function, its joint eigenvalues for sphere averages, and the growth of vector-distance spheres. Conjugation by this function and complex interpolation yield a weighted ℓ^2 estimate and the boundedness of radial operators with ℓ^p kernels for 1≤ p<2. We also present proofs by the averaging operators norms, by the Schur test and by Helgason transform. A further interpolation gives weighted mixed-exponent estimates. No transitivity assumption on the automorphism group is required; in particular the argument applies to buildings of type Ã_2 without a Bruhat--Tits realization.
ER  -

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