[Paper Review] Notes on the spaces of bilinear multipliers

Óscar Blasco|ArXiv.org|May 26, 2009

Advanced Harmonic Analysis Research10 references3 citations

TL;DR

This paper investigates the structure and boundedness properties of bilinear multiplier operators on $ℝ^n$, focusing on multipliers of the form $m(\xi,\eta) = M(\xi - \eta)$, where $M$ is a function on $\mathbb{R}^n$. It establishes sharp $L^p$-boundedness conditions, proves that the multiplier space $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ is trivial under certain index conditions, and provides interpolation-based constructions of nontrivial multipliers via convolution-type operators and Fourier multipliers.

ABSTRACT

A locally integrable function $m(ξ,η)$ defined on $\mathbb R^n imes \mathbb R^n$ is said to be a bilinear multiplier on $\mathbb R^n$ of type $(p_1,p_2, p_3)$ if $$ B_m(f,g)(x)=\int_{\mathbb R^n} \int_{\mathbb R^n}\hat f(ξ)\hat g(η)m(ξ,η)e^{2πi(} dξdη$$ defines a bounded bilinear operator from $L^{p_1}(\mathbb R^n) imes L^{p_2}(\mathbb R^n) $ to $L^{p_3}(\mathbb R^n)$. The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where $m(ξ,η)= M(ξ-η)$ for a given $M$ defined on $\mathbb R^n$ is also addressed.

Motivation & Objective

• To characterize the space of bilinear multipliers $\mathcal{BM}_{(p_1,p_2,p_3)}(\mathbb{R}^n)$ for functions $m(\xi,\eta)$ of the form $M(\xi - \eta)$.
• To determine necessary and sufficient conditions on indices $p_1, p_2, p_3$ for the nontriviality of $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$.
• To develop methods for constructing nontrivial bilinear multipliers using convolution operators and Fourier multiplier techniques.
• To establish sharp $L^p$-boundedness estimates for bilinear operators associated with $M(\xi - \eta)$ via interpolation and duality.

Proposed method

• Introduces the space $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ of functions $M$ such that $m(\xi,\eta) = M(\xi - \eta)$ defines a bounded bilinear multiplier of type $(p_1,p_2,p_3)$.
• Uses the representation $B_M(f,g)(x) = \int \!\!\! \int \hat{f}(\xi)\hat{g}(\eta) M(\xi - \eta) e^{2\pi i \langle \xi + \eta, x \rangle} d\xi d\eta$ to relate multipliers to bilinear convolution operators.
• Applies the theory of $L^p$-boundedness of bilinear operators via the kernel $K$ such that $M = \hat{K}$, linking $M \in \tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ to $K \in L^q(\mathbb{R})$ under index conditions.
• Employs real interpolation and duality arguments to extend boundedness from endpoint cases to intermediate $L^p$ spaces.
• Uses Gaussian regularization and pointwise estimates involving $\lambda \int e^{-\lambda^2 \xi^2} M(\xi + 2y) d\xi$ to prove that $M \equiv 0$ under index conditions $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} > 1$ or $< 1$.
• Applies Young's inequality and Hölder's inequality to control the $L^{p_3}$ norm of $T(K,f,g) = \int f(x-t)g(x+t)K(t) dt$ under $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} = 1 - \frac{1}{q}$.

Experimental results

Research questions

• RQ1 For which indices $p_1, p_2, p_3$ is the space $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ nontrivial?
• RQ2 What is the relationship between the $L^p$-boundedness of bilinear operators with symbol $m(\xi,\eta) = M(\xi - \eta)$ and the integrability of the associated kernel $K$ with $M = \hat{K}$?
• RQ3 Can interpolation techniques be used to construct nontrivial bilinear multipliers in $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$?
• RQ4 Under what conditions does $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R}) = \{0\}$?
• RQ5 How do the $L^p$-bounds of the bilinear operator $B_M(f,g)$ relate to the $L^q$-norm of $K$ when $M = \hat{K}$?

Key findings

• The space $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ is trivial (i.e., contains only the zero function) if $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} > 1$ or $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} < 1$, as shown via Gaussian regularization and pointwise decay of the Fourier transform.
• If $M = \hat{K}$ for $K \in L^q(\mathbb{R})$ with $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} = 1 - \frac{1}{q}$, then $M \in \tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ and $\|M\|_{p_1,p_2,p_3} \leq C\|K\|_q$.
• For $M \in M(\mathbb{R})$, the Fourier transform $\hat{M} = \mu$ satisfies $\hat{M} \in \tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ whenever $\frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{p_3} \leq 1$, with $\|\hat{M}\| \leq \|\mu\|_1$.
• The bilinear operator $T(K,f,g)(x) = \int f(x-t)g(x+t)K(t) dt$ is bounded from $L^{p_1} \times L^{p_2}$ to $L^{p_3}$ under the index condition $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} = 1 - \frac{1}{q}$, with norm controlled by $\|K\|_q$.
• The space $\tilde{\mathcal{M}}_{(p_1,p_2,p_3)}(\mathbb{R})$ is trivial when $\frac{1}{p_1} + \frac{1}{p_2} < \frac{1}{p_3}$ or $\frac{1}{p_1} + \frac{1}{p_2} > \frac{1}{p_3} + 1$, as established by combining Gaussian regularization and the Riemann-Lebesgue lemma.
• Interpolation between $L^1 \times L^{q_1} \times L^{q_2} \to L^s$ and $L^{p'} \times L^{p_1} \times L^{p_2} \to L^\infty$ estimates yields boundedness in $L^q \times L^{p_1} \times L^{p_2} \to L^{p_3}$ for $\frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} = 1 - \frac{1}{q}$.