Endpoint Estimates for Stein's Purely Quadratic Carleson Operator

Abstract

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_λ \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 πi λy^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}.$ We prove that if $\Phi$ is a Young function satisfying $\Phi (t)=o\left(t{\log }_{2}t\right)$ 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 $L({\log }_{2}L{)}^2{\log }_{4}L$ estimate for ${\mathcal{C}}_{2,\mathsf{lac}}.$

arXiv commentNo change in manuscript from v1. Lean formalization of the paper is at https://github.com/anastasiosfragkosmath/lean-verification-quadratic-carleson