[Paper Review] On the Hörmander multiplier theorem and modulation spaces

Naohito Tomita|ArXiv.org|Apr 21, 2008

Mathematical Analysis and Transform Methods16 references3 citations

TL;DR

This paper investigates whether the modulation space $ M^{2,1}_0 $ can replace the Besov space $ B^{2,1}_{n/2} $ in the Hörmander multiplier theorem. It proves that while $ M^{2,1}_s $ with $ s > 0 $ yields $ L^p $-boundedness for Fourier multipliers via interpolation and duality, the critical case $ s = 0 $ fails: there exist multipliers bounded in $ M^{2,1}_0 $ that are not bounded on $ L^p $ for $ p \neq 2 $, showing $ M^{2,1}_0 $ is insufficient to replace $ B^{2,1}_{n/2} $.

ABSTRACT

It is known that the Sobolev space $L^2_s$ with $s>n/2$ appeared in the Hörmander multiplier theorem can be replaced by the Besov space $B^{2,1}_{n/2}$. On the other hand, the Besov space $B_{n/2}^{2,1}$ is continuously embedded in the modulation space $M^{2,1}_0$. In this paper, we consider the problem whether we can replace $B_{n/2}^{2,1}$ by $M^{2,1}_0$.

Motivation & Objective

• Establish whether the modulation space $ M^{2,1}_0 $ can replace the Besov space $ B^{2,1}_{n/2} $ in the Hörmander multiplier theorem.
• Analyze the relationship between $ M^{2,1}_s $, $ B^{2,1}_{n/2} $, and $ L^p $-boundedness of Fourier multipliers for $ 1 < p < ∞ $.
• Investigate the sharpness of the embedding $ B^{2,1}_{n/2} \hookrightarrow M^{2,1}_0 $ and its implications for multiplier theorems.
• Clarify the role of the critical case $ s = 0 $ in $ M^{2,1}_s $-based multiplier conditions.
• Provide a counterexample to show that $ \sup_j \|m_j\|_{M^{2,1}_0} < \infty $ does not imply $ L^p $-boundedness for $ p \neq 2 $.

Proposed method

• The paper uses the decomposition $ m_j(\xi) = \psi(\xi) m(2^j \xi) $, where $ \psi $ is a smooth cut-off supported in an annulus, to localize the multiplier in frequency.
• Key estimates are derived using the norm equivalence $ \|m_j\|_{M^{2,1}_s} \asymp \|(I - \Delta)^{s/2} m_j\|_{\mathcal{F}L^1} $, linking modulation space norms to Fourier image norms.
• Embedding results from Toft and Sugimoto-Tomita are used to compare $ B^{2,1}_{n/2} $ and $ M^{2,1}_0 $, showing no inclusion relation when $ 0 < s < n/2 $.
• Interpolation and duality arguments are applied to extend boundedness from $ H^1 $ to $ L^p $ for $ 1 < p < \infty $, under $ \sup_j \|m_j\|_{M^{2,1}_s} < \infty $ with $ s > 0 $.
• A counterexample from Triebel is adapted to show that $ \sup_j \|m_j\|_{M^{2,1}_0} < \infty $ does not imply $ L^p $-boundedness for $ p \neq 2 $, using properties of $ \mathcal{F}^{-1}m \in B^{1,\infty}_0 $.
• Norm equivalences $ \|m_j\|_{M^{p,1}_s} \asymp \|(I - \Delta)^{s/2} m_j\|_{\mathcal{F}L^1} $ are used to generalize results to $ M^{p,1}_s $ for $ 1 \leq p \leq \infty $.

Experimental results

Research questions

• RQ1 Can the modulation space $ M^{2,1}_0 $ replace the Besov space $ B^{2,1}_{n/2} $ in the Hörmander multiplier theorem?
• RQ2 Is the condition $ \sup_j \|m_j\|_{M^{2,1}_0} < \infty $ sufficient for $ L^p $-boundedness of $ m(D) $ when $ p \neq 2 $?
• RQ3 What is the relationship between $ B^{2,1}_{n/2} $ and $ M^{2,1}_s $ for $ 0 < s < n/2 $, and does this affect multiplier boundedness?
• RQ4 Does the embedding $ B^{2,1}_{n/2} \hookrightarrow M^{2,1}_0 $ imply that $ M^{2,1}_0 $ suffices for the multiplier theorem?
• RQ5 Can the critical case $ s = 0 $ in $ M^{2,1}_s $ be used to characterize $ L^p $-boundedness of Fourier multipliers?

Key findings

• Theorem 1.1 establishes that if $ \sup_j \|m_j\|_{M^{2,1}_s} < \infty $ for some $ s > 0 $, then $ m(D) $ is bounded on $ H^1(\mathbb{R}^n) $, and by interpolation and duality, on $ L^p(\mathbb{R}^n) $ for all $ 1 < p < \infty $.
• Corollary 1.2 shows that the condition $ \sup_j \|\widehat{m_j}\|_{K^{1,1}_s} < \infty $ implies $ H^1 $-boundedness, which is a special case of a result by Baernstein-Sawyer.
• Corollary 1.3 extends the result to $ M^{p,1}_s $ for $ 1 \leq p \leq \infty $, showing that $ \sup_j \|m_j\|_{M^{p,1}_s} < \infty $ with $ s > 0 $ implies $ H^1 $-boundedness.
• Proposition 1.4 provides a negative answer: there exists a multiplier $ m \in \mathcal{S}'(\mathbb{R}^n) $ such that $ \sup_j \|m_j\|_{M^{2,1}_0} < \infty $, but $ m(D) $ is not bounded on $ L^p(\mathbb{R}^n) $ for any $ p \neq 2 $.
• The embedding $ B^{2,1}_{n/2} \hookrightarrow M^{2,1}_0 $ holds, but $ M^{2,1}_0 \not\hookrightarrow B^{2,1}_{n/2} $, and no inclusion exists between $ B^{2,1}_{n/2} $ and $ M^{2,1}_s $ for $ 0 < s < n/2 $, showing the two spaces are not comparable in this range.
• The norm equivalence $ \|m_j\|_{M^{2,1}_s} \asymp \|(I - \Delta)^{s/2} m_j\|_{\mathcal{F}L^1} $ is established via the Fourier image of $ (I - \Delta)^{s/2} m_j $, linking modulation space norms to Fourier restriction estimates.