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Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity

Marco Fraccaroli, Olli Saari, Christoph Thiele

TL;DR

The paper establishes uniform strict-local $L^{2}$ bounds for trilinear Fourier multiplier forms whose singular set is the push-forward of the $d$-diagonal by a block $K$-quasiconformal matrix. The authors develop a high-dimensional phase-space analysis based on a tree/tiles decomposition, introducing core/boundary sizes and a three-stage almost-orthogonality scheme to control paraproduct-type interactions. They reduce the problem to tensorized model forms via a Whitney-type decomposition and a phase-space projection framework drawn from prior work, culminating in a robust, uniform bound that specializes to known results for the bilinear Hilbert transform in 1D and yields new uniform bounds for the bilinear Beurling transform in 2D. The approach systematically handles non-degenerate, partially degenerate, and fully degenerate regimes through a structured decomposition and recursive selection process, enabling summation over trees with controlled amplitudes. Overall, the results advance understanding of uniform bounds for multilinear operators with anisotropic, high-dimensional singularities and have potential implications for related Carleson-type and singular integral problems. $

Abstract

We prove bounds in the strict local $L^{2}(\mathbb{R}^{d})$ range for trilinear Fourier multiplier forms with a $d$-dimensional singular subspace. Given a fixed parameter $K \ge 1$, we treat multipliers with non-degenerate singularity that are push-forwards by $K$-quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local $L^{2}$ range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.

Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity

TL;DR

The paper establishes uniform strict-local bounds for trilinear Fourier multiplier forms whose singular set is the push-forward of the -diagonal by a block -quasiconformal matrix. The authors develop a high-dimensional phase-space analysis based on a tree/tiles decomposition, introducing core/boundary sizes and a three-stage almost-orthogonality scheme to control paraproduct-type interactions. They reduce the problem to tensorized model forms via a Whitney-type decomposition and a phase-space projection framework drawn from prior work, culminating in a robust, uniform bound that specializes to known results for the bilinear Hilbert transform in 1D and yields new uniform bounds for the bilinear Beurling transform in 2D. The approach systematically handles non-degenerate, partially degenerate, and fully degenerate regimes through a structured decomposition and recursive selection process, enabling summation over trees with controlled amplitudes. Overall, the results advance understanding of uniform bounds for multilinear operators with anisotropic, high-dimensional singularities and have potential implications for related Carleson-type and singular integral problems. $

Abstract

We prove bounds in the strict local range for trilinear Fourier multiplier forms with a -dimensional singular subspace. Given a fixed parameter , we treat multipliers with non-degenerate singularity that are push-forwards by -quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range.
Paper Structure (12 sections, 20 theorems, 181 equations)

This paper contains 12 sections, 20 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Proof of Proposition \ref{['globaltreeprop']}. Paraproduct
  4. Proof of Proposition \ref{['thm:tree-estimate']}. Tree estimate
  5. Proof of Proposition \ref{['almostorthogonal']}. Tree selection
  6. Proof of Proposition \ref{['prop:ao-core']}. Core size
  7. Proof of Proposition \ref{['prop:ao-single-lac']}. Boundary size
  8. Proof of Proposition \ref{['prop:ao-sum-lac']}. Sum size
  9. Proof of Proposition \ref{['prop:induction-step']}. Recursion
  10. Conclusion of the proof of Proposition \ref{['almostorthogonal']}
  11. Proof of Proposition \ref{['whitneyprop']}. Tensorized model form
  12. Proof of Theorem \ref{['main-theorem']}

Key Result

Theorem 1.1

Let $d \ge 1$, $K \ge 1$ and There exists a constant $C = C (d,K,p_1,p_2,p_3)$ such that the following holds. Let $m:{\mathbb R}^{3 \times d} \to {\mathbb C}$ satisfy for all $\gamma \in {\mathbb N}^{3 \times d}$ with $\left \lvert \gamma \right \rvert \le 100d$. Let $L$ be a non-trivial block $K$-quasiconformal matrix. Define where $\delta_0$ is the Dirac mass at origin. Then for all tripl

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1: Frequency cut-offs
  • Proposition 2.2: Weak estimate for tensor model
  • Proposition 2.3: Frequency localized estimate
  • Definition 2.4: Multitile, $n$-tile
  • Definition 2.5: Tree
  • Definition 2.6: Main sizes
  • Proposition 2.7: Phase space localized estimate
  • Proposition 2.8: Decomposition of the phase space
  • Proposition 4.1
  • ...and 30 more