Unboundedness Theorems for Symbols Adapted to Large Subspaces

Abstract

For every integer $n\ge 3$, we prove that the n-sublinear generalization of the Bi-Carleson operator of Muscalu, Tao, and Thiele given by nC^{\vecα} :(f_1,..., f_n) \mapsto \sup_{M} \left| \int_{\vecξ \cdot \vecα >0, ξ_n < M} \left[\prod_{j=1}^n \hat{f}_j(ξ_j) e^{2 πi x ξ_j }\right]d\vecξ ~\right|satisfies no $L^p$ estimates provided $\vec{\alpha }\in {\mathbb{Q}}^n$ with distinct, non-zero entries. Furthermore, if $n\ge 5$ and $\vec{\alpha }\in {\mathbb{Q}}^n$ has distinct, non-zero entries, it is shown that there is a symbol $m:{\mathbb{R}}^n\to \mathbb{C}$ 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 Hörmander-Marcinkiewicz symbol $\Pi :{\mathbb{R}}^2\to \mathbb{C}$, which is a paraproduct of $(\varphi ,\psi )$ 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. $\frac{n}{2}+\frac{3}{2}\le \mathfrak{d}<n$ 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-Hörmander 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.