Restricted Carleson Variations at Endpoint and Discretized Hilbert Transforms in the Plane

Abstract

We provide elementary proofs that the 2-variation Carleson operator $V_2$ along with explicit bilinear multipliers adapted to $\{{\xi }_1+{\xi }_2=0\}$ satisfy no $L^p$ estimates. Furthermore, we obtain $L^p\to L^p$ estimates when $2<p<\infty$ for a smooth restricted variant of $V_2$ that is defined a priori on Schwartz functions by the formula \begin{eqnarray*} \mathcal{V}^{res}_2 : f \mapsto \sup_{R \in \mathbb{R}_+} ~~\sup_{0 \leq α< R} ~~\left(\sum_{j \in \mathbb{Z}} \left|f*\mathcal{F}^{-1} \left[ \tilde{1}_{[α+ j R, α+ (j+1)R]}\right] \right|^2 \right)^{1/2} \end{eqnarray*} where ${\tilde{1}}_I(x):=\tilde{1}(|I{|}^{-1}(x-c_I))$ for all intervals $I=[c_I-|I|/2,c_I+|I|/2]\subset \mathbb{R}$ and $\tilde{1}\in C^{\infty }([-1/2,1/2])$. We then study bi-sublinear variants of ${\mathcal{V}}_2^{res}$ before showing that multipliers, which are adapted to $\{{\xi }_1+{\xi }_2=0\}$ and periodically discretized along each frequency scale, map $L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\to L^{p_1{p}_2/(p_1+p_2)}(\mathbb{R})$ provided $2\le p_1,p_2<\infty$ and $\frac{1}{p_1}+\frac{1}{p_2}<1$.

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