[Paper Review] Multilinear Fourier Multipliers with Minimal Sobolev Regularity, II

Loukas Grafakos, Akihiko Miyachi+2 more|arXiv (Cornell University)|Apr 27, 2015

Advanced Harmonic Analysis Research14 references3 citations

TL;DR

This paper establishes sharp $L^p$-boundedness conditions for $m$-linear Fourier multipliers on Hardy and Lebesgue spaces with minimal Sobolev regularity, extending Calderón-Torchinsky and Miyachi-Tomita results to $m \geq 3$. It introduces a novel product-type Sobolev space condition $\|\sigma(2^j\cdot)\widehat{\psi}\|_{W^{(s_1,\dots,s_m)}} < \infty$ and proves that the optimal smoothness $s_i$ depends on the Hardy-Lebesgue exponents and is constant on convex simplices with $m2^{m-1}+1$ vertices, resolving a major technical challenge in multilinear multiplier theory.

ABSTRACT

We provide characterizations for boundedness of multilinear Fourier operators on Hardy-Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0

Motivation & Objective

• To extend the sharp $L^p$-boundedness results for linear and bilinear Fourier multipliers to the multilinear case with $m \geq 3$.
• To characterize the minimal $L^2$-Sobolev regularity required for $m$-linear Fourier multiplier operators to be bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $L^p$.
• To resolve the combinatorial and technical complexity arising in higher-order multilinear settings, particularly when $1 < p_j < 2$.
• To establish that the optimal smoothness condition is constant on convex simplices with $m2^{m-1}+1$ vertices in $[0,\infty)^m$, and to prove boundedness holds throughout the interior of these simplices.

Proposed method

• The authors introduce a product-type Sobolev space $W^{(s_1,\dots,s_m)}$ defined via weighted $L^2$ norms of the Fourier transform with weights $\langle y_i \rangle^{s_i}$.
• They derive boundedness of $m$-linear Fourier multipliers by proving uniform estimates on dyadic annuli using the condition $\sup_{j \in \mathbb{Z}} \|\sigma(2^j \cdot)\widehat{\psi}\|_{W^{(s_1,\dots,s_m)}} < \infty$, where $\widehat{\psi}$ is a smooth partition of unity.
• The proof relies on a multilinear interpolation argument using weak-type estimates at the endpoints of the Lorentz scale, applying a multilinear Calderón-Zygmund decomposition.
• They establish weak-type estimates at the endpoints $p_1 = 1 \pm \epsilon$ and use interpolation to derive the full range of boundedness, with the interpolation parameter $\theta$ determined by $1/p = (1-\theta)/p_0 + \theta/p_1$.
• The key innovation lies in handling the interior of the convex simplex with $m2^{m-1}+1$ vertices by proving estimates uniformly close to the vertices without losing smoothness.
• Necessity of the smoothness conditions is shown via a reduction to known counterexamples in the literature, following the method of Grafakos et al. [14, Theorem 5.1].

Experimental results

Research questions

• RQ1 What is the minimal $L^2$-Sobolev regularity required for an $m$-linear Fourier multiplier operator to be bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $L^p$ for $m \geq 3$?
• RQ2 How does the required smoothness $s_i$ depend on the Hardy-Lebesgue exponents $p_i$ in the multilinear setting, and is it constant across certain geometric regions in the parameter space?
• RQ3 Can the boundedness of $m$-linear multipliers be established uniformly over the interior of a convex simplex with $m2^{m-1}+1$ vertices, even when interpolation between vertices fails to preserve minimal smoothness?
• RQ4 Is the condition $\sup_j \|\sigma(2^j \cdot)\widehat{\psi}\|_{W^{(s_1,\dots,s_m)}} < \infty$ both necessary and sufficient for $m$-linear multiplier boundedness in the full range $0 < p_i \leq \infty$?
• RQ5 How does the multilinear theory generalize the linear results of Calderón and Torchinsky and the bilinear results of Miyachi and Tomita to higher-order multilinear operators?

Key findings

• The optimal $L^p$-boundedness condition for $m$-linear Fourier multipliers from $H^{p_1} \times \cdots \times H^{p_m}$ to $L^p$ is given by $\sup_{j \in \mathbb{Z}} \|\sigma(2^j \cdot)\widehat{\psi}\|_{W^{(s_1,\dots,s_m)}} < \infty$, where $s_i > \frac{n}{p_i} - \frac{n}{2}$ for $p_i \leq 1$ and $s_i > \frac{n}{2}$ for $p_i > 1$, with equality in the exponent being sharp.
• The required smoothness $s_i$ is constant on convex simplices in $[0,\infty)^m$ with $m2^{m-1}+1$ vertices, and boundedness holds throughout the interior of these simplices.
• The authors prove that interpolation between the vertices of the simplex does not preserve minimal smoothness, necessitating a new method to control the interior points uniformly.
• The boundedness result holds even in endpoint cases where some $p_j = \infty$, extending previous results to the full range $0 < p_i \leq \infty$.
• The necessity of the smoothness condition is established via reduction to known counterexamples, confirming that the derived conditions are sharp.
• The method successfully overcomes the combinatorial and technical challenges in the multilinear setting for $m \geq 3$, particularly in the regime $1 < p_j < 2$, where the boundedness region is a non-trivial convex simplex.