Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria

Georgios Dosidis, Bae Jun Park, Lenka Slavikova

TL;DR

This work develops sharp bilinear Fourier multiplier criteria for symbols formed as sums of dyadic dilations of a fixed symbol, with kernels whose sphere restriction lies in $L(\log L)^\alpha$. It establishes a precise threshold on the log-regularity parameter $\lambda$ for linear and bilinear boundedness and then applies these results to rough bilinear singular integrals, including kernels associated with $\Omega$ on the sphere in $L(\log L)^A$. In particular, it shows that for $\Omega\in L(\log L)^A$ with $A$ large enough (and, in the limiting endpoint, $A\ge 2$), the bilinear operator is bounded on $H^{p_1}\times H^{p_2}$ to $Y^p$, with sharp necessity conditions on the exponents. The methods blend shifted square function estimates, vector-valued Hardy/BMO theory, and dyadic decomposition to bridge linear endpoint theories to the bilinear and multilinear setting, yielding near-endpoint results and a flexible framework for rough multilinear operators.

Abstract

We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space $L(\log L)^α$. This improves the previously best known condition for boundedness of such bilinear operators obtained in the paper of the first and third authors, and provides estimates close to the conjectured endpoint of integrability suggested by the linear theory. The proof is based on a new sharp boundedness criterion for bilinear Fourier multiplier operators associated with sums of dyadic dilations of a fixed symbol $m_0$, compactly supported away from the origin. This criterion admits the best possible behavior with respect to a modulation of $m_0$ and is intimately connected with sharp shifted square function estimates.

Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria

TL;DR

This work develops sharp bilinear Fourier multiplier criteria for symbols formed as sums of dyadic dilations of a fixed symbol, with kernels whose sphere restriction lies in . It establishes a precise threshold on the log-regularity parameter for linear and bilinear boundedness and then applies these results to rough bilinear singular integrals, including kernels associated with on the sphere in . In particular, it shows that for with large enough (and, in the limiting endpoint, ), the bilinear operator is bounded on to , with sharp necessity conditions on the exponents. The methods blend shifted square function estimates, vector-valued Hardy/BMO theory, and dyadic decomposition to bridge linear endpoint theories to the bilinear and multilinear setting, yielding near-endpoint results and a flexible framework for rough multilinear operators.

Abstract

We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space . This improves the previously best known condition for boundedness of such bilinear operators obtained in the paper of the first and third authors, and provides estimates close to the conjectured endpoint of integrability suggested by the linear theory. The proof is based on a new sharp boundedness criterion for bilinear Fourier multiplier operators associated with sums of dyadic dilations of a fixed symbol , compactly supported away from the origin. This criterion admits the best possible behavior with respect to a modulation of and is intimately connected with sharp shifted square function estimates.
Paper Structure (16 sections, 6 theorems, 177 equations, 2 figures)

This paper contains 16 sections, 6 theorems, 177 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardy spaces and $BMO$
  4. Maximal inequalities
  5. A vector-valued characterization of $H^p$ and $BMO$
  6. Shifted square function estimates
  7. Proof of Proposition \ref{['T:Hormander']}
  8. Proof of \ref{['E:opt-est']}
  9. Optimality of the condition $\lambda \geq |1/2-1/p|$
  10. Proof of Theorem \ref{['T:HormanderB']}
  11. Proof of \ref{['E:BoundB']}
  12. Optimality of the condition $\lambda\ge \max\{\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p'}\}$
  13. Proof of Theorem \ref{['T:bilinearresult']}
  14. Decomposition of the kernel
  15. The case $j>0$
  16. ...and 1 more sections

Key Result

Proposition 1

Let $1\le p\le \infty$ and $\lambda \geq 0$. Let $K$ be an integrable function on ${\mathbb R}^n$ whose Fourier transform is supported in the set $\{\xi\in {\mathbb R}^n:~1/2 \leq |\xi| \leq 2\}$ and which satisfies Let $T$ be the linear operator associated with $K$, defined by E:T. If $\lambda \ge |1/2-1/p|$, then there is a constant $C = C(n,p)$, such that for all $f\in\mathscr{S}_0(\mathbb R^

Figures (2)

  • Figure 1: Range of $p_1,p_2$ for $H^{p_1}\times H^{p_2}\to Y^p$ boundedness of $\mathcal{B}$ for a given value of $\lambda$.
  • Figure 2: Range of $p_1,p_2$ for $H^{p_1}\times H^{p_2}\to Y^p$ boundedness of $\mathcal{B}$ for a given value of $\lambda$, split into three regions depending on which of the numbers $\frac{1}{p_1}$, $\frac{1}{p_2}$ and $\frac{1}{p'}$ is the largest.

Theorems & Definitions (9)

  • Proposition 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Lemma 4
  • Corollary 5
  • Lemma 6
  • proof
  • proof : Proof of Lemma \ref{['lpbmoest']}