PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Unboundedness theorems for symbols adapted to large subspaces

Volume 244 / 2019

Robert Kesler Studia Mathematica 244 (2019), 109-158 MSC: Primary 42B15; Secondary 42B20. DOI: 10.4064/sm8750-7-2017 Published online: 16 May 2018


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.


  • Robert KeslerSchool of Mathematics
    Georgia Tech
    Atlanta, GA 30332, U.S.A.

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image