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

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 $L^p \rightarrow L^p$ 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 \alpha < R} ~~\left(\sum_{j \in \mathbb{Z}} \left|f*\mathcal{F}^{-1} \left[ \tilde{1}_{[\alpha + j R, \alpha + (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 $L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \rightarrow L^{p_1 p_2 / (p_1 + p_2)}(\mathbb{R})$ provided $2 \leq p_1, p_2 <\infty$ and $\frac{1}{p_1} + \frac{1}{p_2} <1$.