Endpoint Estimates for Stein's Purely Quadratic Carleson Operator

Abstract: We study the near $L<sup>1$ behavior of the maximally quadratically modulated Hilbert transform [ \mathcal{C}2f(x) := \supλ \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 πi λy^2} \frac{\mathrm{d} y }{y} \right| ] and its lacunary counterpart obtained by restricting $λ$ to $2<sup>{\mathbb{Z}}.$ We prove that if $Φ$ is a Young function satisfying $Φ(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}<em>{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ 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 $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)² \log_4L$ estimate for $\mathcal{C}</em>{2,\mathsf{lac}}.$