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Boundedness of bilinear radial Fourier multipliers

Petr Honzík, Matyáš Maleček

Abstract

We show that a bilinear radial Fourier multiplier operator with symbol $σ$ is $L^2(\R^n)\times L^2(\R^n) \to L^1(\R^n)$ bounded, $n\in \mathbb N,$ if the function $σ$ satisfies the smoothness condition $σ(2^j\cdot)Φ\in L^2_{1/2 +ε}(\mathbb R^{2n})$ for some $ε>0$ and every $j\in \mathbb Z,$ where $Φ$ is a smooth cutoff function adapted to the annulus $|x|\in [1/4,4]$. This condition is dimension free. We also apply similar reasoning to provide alternative proof of the initial result concerning multilinear Bochner-Riesz operator and prove an estimate for generalized bilinear Bochner-Riesz operator.

Boundedness of bilinear radial Fourier multipliers

Abstract

We show that a bilinear radial Fourier multiplier operator with symbol is bounded, if the function satisfies the smoothness condition for some and every where is a smooth cutoff function adapted to the annulus . This condition is dimension free. We also apply similar reasoning to provide alternative proof of the initial result concerning multilinear Bochner-Riesz operator and prove an estimate for generalized bilinear Bochner-Riesz operator.
Paper Structure (11 sections, 9 theorems, 45 equations)

This paper contains 11 sections, 9 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Definitions and lemmas
  3. Proof of Lemma \ref{['abs_convergence_lemma']}
  4. Proof of key Lemma \ref{['key_lemma']}
  5. Proof of Lemma \ref{['main_lemma']}
  6. Diagonal part
  7. Off-diagonal parts:
  8. Proof of theorem \ref{['Bochner_Riesz']}
  9. Boundedness of $T_{\sigma_1}$
  10. Boundedness of $T_{\sigma_2}$
  11. Proof of theorem \ref{['Modified_Bochner_Riesz']}

Key Result

Theorem 1

Suppose $\widehat{\psi}\in\mathcal{C}_0^{\infty}(\mathbb R^{2n})$ is positive and supported in the annulus such that $\sum_{j\in\mathbb Z}\widehat{\psi}_j(\xi,\eta)= \sum_{j}\widehat{\psi}(2^{-j}(\xi,\eta))=1$ for all $(\xi,\eta)\neq0$. Let $1< r<\infty$, $s>\max\{n/2, 2n/r\}$, and suppose there is a constant $A$ such that Then there is a constant $C=C(n,\Psi)$ such that the bilinear operator i

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9