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Improved sup-norm bounds for locally symmetric spaces

  • Christopher Lutsko
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Abstract

Let X=G/K be a symmetric space of noncompact type, of dimension n and rank r, and let Y=Γbackslash X. Sarnak's local bound for an L2-normalized spherical joint eigenfunction with regular tempered parameter of size T is |ϕ|∞≪ T(n−r)/2. We prove o(T(n−r)/2) locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range y≤ Tβ, β<1/2. If the injectivity radius is bounded below, we prove the global estimate T(n−r)/2(log T)−r/2. The proof makes use of a novel kernel argument and a uniform bound on the spherical function.

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Cite

BibTeXRIS
BibTeX
@misc{lutsko2026improved,
  title = {{Improved sup-norm bounds for locally symmetric spaces}},
  author = {Christopher Lutsko},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.21580},
  eprint = {2608.21580},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21580},
}
RIS
TY  - UNPB
TI  - Improved sup-norm bounds for locally symmetric spaces
AU  - Christopher Lutsko
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2608.21580
UR  - https://arxiv.org/abs/2608.21580
AB  - Let X=G/K be a symmetric space of noncompact type, of dimension n and rank r, and let Y=Γbackslash X. Sarnak's local bound for an L^2-normalized spherical joint eigenfunction with regular tempered parameter of size T is |ϕ|_∞≪ T^(n−r)/2. We prove o(T^(n−r)/2) locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range y≤ T^β, β<1/2. If the injectivity radius is bounded below, we prove the global estimate T^(n−r)/2(log T)^−r/2. The proof makes use of a novel kernel argument and a uniform bound on the spherical function.
ER  -