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^{\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 $\vecα \in \mathbb{Q}^n$ with distinct, non-zero entries. Furthermore, if $n \geq 5$ and $\vecα \in \mathbb{Q}^n$ has distinct, non-zero entries, it is shown that there is a symbol $m:\mathbb{R}^n \rightarrow \mathbb{C}$ adapted to the hyperplane $Γ^{\vec{a}}=\left\{ \vecξ \in \mathbb{R}^n: \sum_{j=1}^n ξ_j \cdot a_j =0 \right\} $ and supported in $\left\{ \vecξ : dist(\vecξ, Γ^{\vecα}) \lesssim 1 \right\}$ for which the associated $n$-linear multiplier also satisfies no $L^p$ estimates. Next, we construct a Hörmander-Marcinkiewicz symbol $Π: \mathbb{R}^2 \rightarrow \mathbb{C}$, which is a paraproduct of $(ϕ, ψ)$ type, such that the trilinear operator $T_m$ whose symbol $m$ is $ sgn(ξ_1 + ξ_2) Π(ξ_2, ξ_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 $Γ\in \mathcal{C}$ there is an associated symbol $m_Γ$ adapted to $Γ$ in the Mikhlin-Hörmander sense and supported in $\left\{ \vecξ : dist(\vecξ, Γ) \lesssim 1 \right\}$ for which the associated multilinear multiplier $T_{m_Γ}$ is unbounded.