Unboundedness theorems for symbols adapted to large subspaces
Volume 244 / 2019
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 $$ C_{\vec{\alpha}} :(f_1,\ldots, f_n) \mapsto \sup_{M \in \mathbb{R}} \Big| \int_{\substack{\vec{\xi} \cdot \vec{\alpha} \gt 0 \\ \xi_n \lt M}} \Big[\prod_{j=1}^n \hat{f}_j(\xi_j) e^{2 \pi i x \xi_j }\Big]d\vec{\xi} \Big| $$ 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{\alpha}}:=\{ \vec{\xi} \in \mathbb{R}^n: \sum_{j=1}^n \xi_j \cdot \alpha_j =0\} $ and supported in $\{ \vec{\xi} : \operatorname{dist}(\vec{\xi}, \Gamma^{\vec{\alpha}}) \leq 1\}$ for which the associated $n$-linear multiplier also satisfies no $L^p$ estimates. Next, we construct a 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 $ \operatorname{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 $(\mathfrak{d},n) $ of integers such that $ {n}/{2}+{3}/{2} \leq \mathfrak{d} \lt 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 $\Gamma \in \mathfrak{C}$ there is an associated symbol $m_\Gamma$ adapted to $\Gamma$ and supported in $\{ \vec{\xi} : \operatorname{dist}(\vec{\xi}, \Gamma) \leq 1\}$ for which the associated multilinear multiplier $T_{m_\Gamma}$ is unbounded.
