Unboundedness Theorems for Symbols Adapted to Large Subspaces

Abstract

For every integer $n \geq 3$, we prove that the n-sublinear generalization of the Bi-Carleson operator of Muscalu, Tao, and Thiele given by nC^{\vec{\alpha}} :(f_1,..., f_n) \mapsto \sup_{M} \left| \int_{\vec{\xi} \cdot \vec{\alpha} >0, \xi_n < M} \left[\prod_{j=1}^n \hat{f}_j(\xi_j) e^{2 \pi i x \xi_j }\right]d\vec{\xi} ~\right|satisfies no $L^p$ estimates provided $\vec{\alpha} \in \mathbb{Q}^n$ with distinct, non-zero entries. Furthermore, if $n \geq 5$ and $\vec{\alpha} \in \mathbb{Q}^n$ has distinct, non-zero entries, it is shown that there is a symbol $m:\mathbb{R}^n \rightarrow \mathbb{C}$ adapted to the hyperplane $\Gamma^{\vec{a}}=\left\{ \vec{\xi} \in \mathbb{R}^n: \sum_{j=1}^n \xi_j \cdot a_j =0 \right\} $ and supported in $\left\{ \vec{\xi} : dist(\vec{\xi}, \Gamma^{\vec{\alpha}}) \lesssim 1 \right\}$ for which the associated $n$-linear multiplier also satisfies no $L^p$ estimates. Next, we construct a H\"{o}rmander-Marcinkiewicz symbol $\Pi: \mathbb{R}^2 \rightarrow \mathbb{C}$, which is a paraproduct of $(\phi, \psi)$ type, such that the trilinear operator $T_m$ whose symbol $m$ is $ sgn(\xi_1 + \xi_2) \Pi(\xi_2, \xi_3)$ satisfies no $L^p$ estimates. Finally, we state a converse to a theorem of Muscalu, Tao, and Thiele using Riesz kernels in the spirit of Muscalu's recent work: for every pair of integers $(\mathfrak{d},n) $ s.t. $ \frac{n}{2}+\frac{3}{2} \leq \mathfrak{d}<n$ there is an explicit collection $\mathfrak{C}$ of uncountably many $\mathfrak{d}$-dimensional non-degenerate subspaces of $\mathbb{R}^n$ such that for each $\Gamma \in \mathcal{C}$ there is an associated symbol $m_\Gamma$ adapted to $\Gamma$ in the Mikhlin-H\"{o}rmander sense and supported in $\left\{ \vec{\xi} : dist(\vec{\xi}, \Gamma) \lesssim 1 \right\}$ for which the associated multilinear multiplier $T_{m_\Gamma}$ is unbounded.