Spherical Averages on nilpotent Groups

Abstract

In this paper, we study the spherical maximal operator ℳ_S^d−1(A), which is associated with the sphere S^d−1 embedded in the horizontal plane of some nilpotent group 𝔾_A. Here, 𝔾_A is the group identified with ℝ^d+1, and its group law is determined by the d× d real matrix A. Specifically, when A is a skew-symmetric matrix J, which is the case of the Heisenberg group, it was known that the spherical maximal operator ℳ_S^d−1(J) is bounded on L^p(𝔾_J) for p in the range (d/(d−1),∞]. In this paper, we extend the results from the skew-symmetric matrix J to a more general class of invertible matrices A. We aim to classify the range of p for which the operator ℳ_S^d−1(A) is bounded on L^p(𝔾_A) based on the eigenvalues of A.

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BibTeX RIS

BibTeX

@misc{kim2026spherical,
  title = {{Spherical Averages on nilpotent Groups}},
  author = {Joonil Kim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36828},
  eprint = {2610.36828},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36828},
}

RIS

TY  - UNPB
TI  - Spherical Averages on nilpotent Groups
AU  - Joonil Kim
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36828
UR  - https://arxiv.org/abs/2610.36828
AB  - In this paper, we study the spherical maximal operator ℳ_S^d−1(A), which is associated with the sphere S^d−1 embedded in the horizontal plane of some nilpotent group 𝔾_A. Here, 𝔾_A is the group identified with ℝ^d+1, and its group law is determined by the d× d real matrix A. Specifically, when A is a skew-symmetric matrix J, which is the case of the Heisenberg group, it was known that the spherical maximal operator ℳ_S^d−1(J) is bounded on L^p(𝔾_J) for p in the range (d/(d−1),∞]. In this paper, we extend the results from the skew-symmetric matrix J to a more general class of invertible matrices A. We aim to classify the range of p for which the operator ℳ_S^d−1(A) is bounded on L^p(𝔾_A) based on the eigenvalues of A.
ER  -

Abstract

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