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Endpoint Estimates for Stein's Purely Quadratic Carleson Operator
Abstract
We study the near L1 behavior of the maximally quadratically modulated Hilbert transform 𝒞2f(x) := supλ |p.v. ∫ℝf(x−y)e2 πi λy2 (d y )/y | and its lacunary counterpart obtained by restricting λ to 2ℤ. We prove that if Φ is a Young function satisfying $ Φ(t)=o\bigl(t\log_2t\bigr)$ then 𝒞2,lac, and therefore 𝒞2 as well, does not satisfy a corresponding Φ-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type (1,1). In the positive direction, we establish an Llog1 L modular estimate for 𝒞2 and a L (log2L)2 log4L estimate for 𝒞2,lac.
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BibTeX
@misc{fragkos2026endpoint,
title = {{Endpoint Estimates for Stein's Purely Quadratic Carleson Operator}},
author = {Anastasios Fragkos and Ben Krause and Michael Lacey},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04101},
eprint = {2609.04101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04101},
}
RIS
TY - UNPB
TI - Endpoint Estimates for Stein's Purely Quadratic Carleson Operator
AU - Anastasios Fragkos
AU - Ben Krause
AU - Michael Lacey
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.04101
UR - https://arxiv.org/abs/2609.04101
AB - We study the near L^1 behavior of the maximally quadratically modulated Hilbert transform 𝒞_2f(x) := sup_λ |p.v. ∫_ℝf(x−y)e^2 πi λy^2 (d y )/y | and its lacunary counterpart obtained by restricting λ to 2^ℤ. We prove that if Φ is a Young function satisfying $ Φ(t)=o\bigl(t\log_2t\bigr)$ then 𝒞_2,lac, and therefore 𝒞_2 as well, does not satisfy a corresponding Φ-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type (1,1). In the positive direction, we establish an Llog_1 L modular estimate for 𝒞_2 and a L (log_2L)^2 log_4L estimate for 𝒞_2,lac.
ER -