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arXiv · 2609.04101

Endpoint Estimates for Stein's Purely Quadratic Carleson Operator

Anastasios Fragkos·Ben Krause·Michael Lacey
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Abstract

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}.$ We prove that if $\Phi$ is a Young function satisfying $ \Phi(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}_{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ as well, does not satisfy a corresponding $\Phi$-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 $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)^2 \log_4L$ estimate for $\mathcal{C}_{2,\mathsf{lac}}.$

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BibTeXRIS

Anastasios Fragkos, Ben Krause, Michael Lacey. 2026-09-03. Endpoint Estimates for Stein's Purely Quadratic Carleson Operator. https://arxiv.org/abs/2609.04101

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