Endpoint Estimates for Stein's Purely Quadratic Carleson Operator

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform $${\mathcal{C}}_{2}f(x):=\underset{\lambda }{\sup }\mid \mathrm{p.v.}{\int }_{\mathbb{R}}f(x-y)e^{2\pi i\lambda y^2}\frac{\mathrm{d}y}{y}\mid$$ and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}\mathrm{.}$ We prove that if $\mathrm{\Phi }$ is a Young function satisfying $\mathrm{\Phi }(t)=o\left(t{\log }_{2}t\right)$ then ${\mathcal{C}}_{2,\mathsf{l}\mathsf{a}\mathsf{c}}$, and therefore ${\mathcal{C}}_2$ as well, does not satisfy a corresponding $\mathrm{\Phi }$-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 $L({\log }_{2}L{)}^2{\log }_{4}L$ estimate for ${\mathcal{C}}_{2,\mathsf{l}\mathsf{a}\mathsf{c}}\mathrm{.}$