[Paper Review] Multilinear Multiplier Theorems and Applications

Loukas Grafakos, Danqing He+2 more|arXiv (Cornell University)|Nov 25, 2015

Advanced Harmonic Analysis Research3 citations

TL;DR

This paper establishes new multilinear multiplier theorems for symbols with limited smoothness, specifically those satisfying a coordinate-wise derivative condition where each variable's derivative order is bounded by $ s_j/n $. The authors prove boundedness of multilinear operators under these weakened smoothness conditions and apply the results to establish sharp $ L^p $ boundedness for Calderón and Calderón-Coifman-Journé commutators in the full range $ p > 1/2 $, extending prior endpoint results via a novel Sobolev space framework on $ \mathbb{R}^{2m} $.

ABSTRACT

We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calderón and Calderón-Coifman-Journé.

Motivation & Objective

• Address the lack of multilinear multiplier theorems for symbols with restricted smoothness, particularly when derivatives are bounded per coordinate.
• Extend Hörmander-type multiplier theorems to the multilinear setting with $ L^r $-based Sobolev norms in a coordinate-wise manner.
• Provide a direct proof of the boundedness of Calderón and Calderón-Coifman-Journé commutators in the full range $ p > 1/2 $, including endpoint cases.
• Establish a framework for handling multilinear operators with symbols that decay slowly but satisfy derivative constraints per variable.
• Bridge the gap between classical Coifman-Meyer multipliers and commutator operators by weakening smoothness assumptions using localized Sobolev norms.

Proposed method

• The paper introduces a coordinate-wise multilinear multiplier condition where each partial derivative $ \partial_{\xi_{ji}}^{\beta_{j\ell}} $ of the symbol $ \sigma $ satisfies decay $ \lesssim (|\xi_1| + \cdots + |\xi_m|)^{-\sum \beta_{j\ell}} $, with $ \beta_{j\ell} \leq s_j/n $.
• Define a mixed-norm Sobolev space $ L^r_{\gamma} $ on $ \mathbb{R}^{2m} $, where the norm involves local $ L^r $-integrability of $ (I - \partial_{\xi_{ji}}^2)^{\gamma_{ji}/2} \sigma $, with $ \gamma_{ji} \leq s_j/n $.
• Use dyadic frequency localization via a bump function $ \widehat{\Psi} $ supported in an annulus to decompose the symbol and control $ \| \sigma(2^k \cdot) \widehat{\Psi} \|_{L^r_{\gamma}} $ uniformly in $ k $.
• Apply a multilinear interpolation argument based on the $ L^r_{\gamma} $-norm condition to deduce boundedness of the associated multilinear operator $ T_\sigma $ from $ L^{p_1} \times \cdots \times L^{p_m} $ to $ L^p $.
• Apply the main multiplier theorem to the bilinear multiplier $ m(\xi; \eta) = \prod_{i=1}^n \textup{sgn}(\eta_i) \Phi(\xi_i / \eta_i) $, arising from the Calderón commutator, to prove $ L^p $ boundedness.
• Use the fact that $ \sigma(2^k \cdot) \widehat{\Psi} \in L^r_{\gamma/2, \gamma/2}(\mathbb{R}^2) $ uniformly in $ k $, and apply Corollary 1.5 to extend boundedness to $ n $-dimensional commutators.

Experimental results

Research questions

• RQ1 How can multilinear multiplier theorems be extended to symbols with limited smoothness, particularly when derivatives are restricted per coordinate variable?
• RQ2 What is the minimal smoothness condition on a multilinear multiplier symbol that still ensures boundedness of the associated operator on $ L^p $ spaces?
• RQ3 Can the boundedness of Calderón commutators be established directly using a multilinear multiplier framework with $ L^r $-Sobolev norms?
• RQ4 Does the coordinate-wise derivative condition $ \beta_{j\ell} \leq s_j/n $ allow for a weaker smoothness assumption than the classical Coifman-Meyer condition?
• RQ5 Can the full range $ p > 1/2 $ for Calderón-Coifman-Journé commutators be recovered via a multilinear multiplier approach without time-frequency analysis?

Key findings

• The paper establishes a new multilinear multiplier theorem for symbols $ \sigma $ satisfying a coordinate-wise derivative decay condition: $ \big| \partial_{\xi_{j1}}^{\beta_{j1}} \cdots \partial_{\xi_{jn}}^{\beta_{jn}} \sigma(\xi_1, \dots, \xi_m) \big| \lesssim (|\xi_1| + \cdots + |\xi_m|)^{-\sum \beta_{j\ell}} $, with $ \beta_{j\ell} \leq s_j/n $.
• Under the condition that $ \sup_{k \in \mathbb{Z}} \| \sigma(2^k \cdot) \widehat{\Psi} \|_{L^r_{\gamma}(\mathbb{R}^{2m})} < \infty $ for $ r, \gamma > 1 $, the associated multilinear operator $ T_\sigma $ is bounded from $ L^{p_1} \times \cdots \times L^{p_m} $ to $ L^p $ for $ 1 < p_j < \infty $, $ 1/p = \sum 1/p_j $, and $ 1/2 < p < \infty $.
• Proposition 6.6 proves that the Calderón commutator $ \mathcal{C}_1 $ is bounded from $ L^{p_1} \times L^{p_2} $ to $ L^p $ for $ 1/2 < p < \infty $, $ 1/p_1 + 1/p_2 = 1/p $, via the new multiplier theorem.
• Proposition 6.7 establishes the boundedness of the $ n $-dimensional Calderón-Coifman-Journé commutator $ \mathcal{C}_1^{(n)} $ from $ L^{p_1} \times L^{p_2} $ to $ L^p $ for $ 1/2 < p < \infty $, with the same exponent conditions.
• The proof relies on decomposing the symbol $ m(\xi; \eta) = \prod_{i=1}^n \textup{sgn}(\eta_i) \Phi(\xi_i / \eta_i) $ and verifying that $ \sigma(2^k \cdot) \widehat{\Psi} \in L^r_{\gamma/2, \gamma/2}(\mathbb{R}^2) $ uniformly in $ k $, which implies the required $ L^r_{\gamma} $-norm condition.
• By Corollary 1.5, the boundedness of $ \mathcal{C}_1^{(n)} $ follows directly from the uniform $ L^r_{\gamma/2, \gamma/2} $-norm control, providing a new, direct proof independent of time-frequency analysis.