Unboundedness Theorems for Symbols Adapted to Large Subspaces

TL;DR

This paper demonstrates that certain multilinear operators and multipliers, even when adapted to specific subspaces or symbols, do not satisfy any $L^p$ estimates, revealing fundamental unboundedness in these advanced harmonic analysis constructs.

Contribution

The paper establishes new unboundedness results for multilinear operators and multipliers adapted to subspaces and symbols, extending previous work and providing explicit counterexamples.

Findings

No $L^p$ estimates for the n-sublinear generalization of the Bi-Carleson operator when $oldsymbol{oldsymbol{eta} eq 0}$.

Existence of symbols adapted to hyperplanes with no $L^p$ bounds for certain multilinear multipliers.

Construction of specific symbols and operators that fail to satisfy $L^p$ estimates, including paraproducts and Riesz kernel-based multipliers.

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.