Homogeneous Polynomial Carleson Operators Along the Paraboloid

Abstract

In this paper, we prove boundedness on L^p(ℝⁿ⁺¹), 1<p<∞, of polynomial Carleson operators along the paraboloid with phase belonging to a class Y of homogeneous polynomials. For n,d≥ 2, let 𝒫^n,d denote the class of all real-valued homogeneous polynomials in n variables of degree d. We may take the class Y to be an arbitrary linear subspace Y ⊆ 𝒫^n,d such that dim Y ≤ M_{n , d}, where M_{n , d} is a constant depending only on n and d. The main novelty of our result is that we may allow a large collection of homogeneous polynomials of the same degree, in contrast to earlier results that relied on distinct degree assumptions. The heart of our proof is a TT^* method, and the derivation of new estimates for oscillatory integral kernels involving interactions between homogeneous polynomial phases and the paraboloid.

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BibTeX RIS

BibTeX

@misc{zhu2026homogeneous,
  title = {{Homogeneous Polynomial Carleson Operators Along the Paraboloid}},
  author = {Alina Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05931},
  eprint = {2611.05931},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05931},
}

RIS

TY  - UNPB
TI  - Homogeneous Polynomial Carleson Operators Along the Paraboloid
AU  - Alina Zhu
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05931
UR  - https://arxiv.org/abs/2611.05931
AB  - In this paper, we prove boundedness on L^p(ℝ^n+1), 1<p<∞, of polynomial Carleson operators along the paraboloid with phase belonging to a class Y of homogeneous polynomials. For n,d≥ 2, let 𝒫^n,d denote the class of all real-valued homogeneous polynomials in n variables of degree d. We may take the class Y to be an arbitrary linear subspace Y ⊆ 𝒫^n,d such that dim Y ≤ M_n , d, where M_n , d is a constant depending only on n and d. The main novelty of our result is that we may allow a large collection of homogeneous polynomials of the same degree, in contrast to earlier results that relied on distinct degree assumptions. The heart of our proof is a TT^* method, and the derivation of new estimates for oscillatory integral kernels involving interactions between homogeneous polynomial phases and the paraboloid.
ER  -

Abstract

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