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Endpoint Criteria for One-Dimensional Bilinear Rough Singular Integrals

This paper establishes endpoint boundedness theorems for one-dimensional bilinear rough singular integrals by proving that the associated angular multiplier has bounded variation if and only if the antipodal even part of the kernel belongs to $H^1(\mathbb{S}^1)$ , thereby deriving boundedness results for kernels in $L\log L$ and under critical directional assumptions.

Original authors: Binwei Dan, Qingying Xue

Published 2026-07-21

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CC BY 4.0

Original authors: Binwei Dan, Qingying Xue

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of mathematics as a giant, bustling kitchen where chefs (mathematicians) are trying to bake the perfect cake. But instead of flour and sugar, they are mixing abstract ingredients like "functions" and "integrals." Sometimes, these ingredients are messy, rough, or "rough" in a mathematical sense—meaning they have jagged edges, sudden jumps, or weird spikes that make them hard to work with. The goal is to figure out exactly how rough an ingredient can be before the whole cake collapses. This field is called harmonic analysis, and it's the study of how waves and signals behave when you chop them up, mix them, or transform them.

In this specific story, the chefs are dealing with a "bilinear" recipe. That's just a fancy way of saying they are mixing two different ingredients (let's call them Ingredient A and Ingredient B) together to create a new result. The trouble comes from a "rough kernel," which is like a spice that has been ground up so poorly it has sharp, jagged bits. If you use too much of this jagged spice, or if it's too unevenly distributed, the mixing process (the integral) might blow up and give you an infinite, nonsensical result. The big question mathematicians have been asking for decades is: What is the absolute limit of how rough this spice can be before the recipe fails? Is there a specific "roughness threshold" that guarantees the cake will still turn out delicious (mathematically, "bounded")?

This paper by Binwei Dan and Qingying Xue dives right into that kitchen to find the exact tipping point for a one-dimensional version of this problem. They discovered that there isn't just one single rule for how rough the spice can be; instead, there are two completely different "flavors" of roughness that work, and they don't overlap.

First, they looked at the "global" roughness. Imagine the spice is a pile of sand where some grains are huge boulders and others are fine dust. They proved that if the spice belongs to a specific category called $L \log L$ , the recipe works perfectly. Think of $L \log L$ as a special club for spices that are mostly fine but allow for a few very large, jagged boulders, as long as the total amount of jaggedness doesn't get out of hand. The authors showed that the "log" part of this name is the exact, non-negotiable limit. If you try to make the spice slightly less rough (removing that "log" factor), the cake collapses. They proved this by showing that if you try to use a spice that is just a tiny bit rougher than this limit, the mathematical mixing machine breaks down.

Second, they investigated a different kind of roughness called "directional" roughness. Imagine the spice isn't just a pile, but a wind that blows harder in some directions than others. The authors found that if the spice is rough in a specific way related to how it behaves when you look at it from different angles (specifically, if it has a certain logarithmic correction near the "critical" angle), the recipe also works. This is like saying the spice is allowed to be very jagged, but only if those jagged bits are arranged in a very specific, balanced pattern around the circle.

Here is the most surprising part of their discovery: these two rules are incomparable. It's like saying "Rule A" is about how heavy your backpack is, and "Rule B" is about how tall you are. You can have a heavy backpack and be short, or a light backpack and be tall. There is no rule that says "If you follow Rule A, you automatically follow Rule B." The authors built specific examples of spices that fit Rule A but break Rule B, and other spices that fit Rule B but break Rule A. This means there is no single "master rule" for roughness; you have to check which specific type of roughness you are dealing with.

The paper doesn't just guess these limits; it proves them with rigorous mathematical logic. They used a clever trick involving "rotations" (spinning the problem around) to turn the messy spice into a smoother profile, and they used "wavelets" (mathematical microscopes that look at signals at different zoom levels) to break the problem into tiny, manageable pieces. They showed that if you stay within these two specific boundaries, the bilinear singular integral operator remains "bounded"—meaning it produces a finite, sensible result no matter what valid ingredients you throw at it. But step outside these boundaries, and the result becomes infinite. It's a definitive map of the safe zone for these rough mathematical recipes, showing exactly where the edge of the cliff is.

Technical Summary: Endpoint Criteria for One-Dimensional Bilinear Rough Singular Integrals

Problem Statement
This paper addresses the endpoint theory for the one-dimensional bilinear rough singular integral operator $T_\Omega$ , defined by:
$T_\Omega(f, g)(x) := \text{p.v.} \int_{\mathbb{R}^2} \frac{\Omega(y/|y|)}{|y|^2} f(x-y_1)g(x-y_2) \, dy_1 dy_2,$
where $\Omega$ is a mean-zero function on the unit circle $S^1$ with no imposed smoothness. The central problem is to determine the minimal angular regularity required on $\Omega$ to ensure the boundedness of $T_\Omega$ from $L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R})$ to $L^p(\mathbb{R})$ for the finite-exponent Banach range ( $1 < p_1, p_2, p < \infty$ , $1/p = 1/p_1 + 1/p_2$ ).

While previous results (e.g., He–Park, Dosidis–Slavíková) established boundedness for $\Omega \in L^q(S^1)$ with $q > 1$ , and Orlicz-type results (Dosidis–Park–Slavíková) provided bounds for $\Omega \in L(\log L)^A$ with $A \geq 1 + \max\{1/p_1, 1/p_2, 1/p'\}$ , the exact endpoint threshold for $A$ and the structural nature of the one-dimensional endpoint remained open. Specifically, it was unclear whether the logarithmic exponent $A=1$ (i.e., $\Omega \in L \log L$ ) suffices and how this compares to directional kernel conditions.

Methodology
The authors employ two distinct analytical frameworks to derive endpoint criteria, corresponding to two different types of kernel regularity:

Rotational Approach and BV Characterization:
The authors utilize the method of rotations to reduce the bilinear operator to an angular multiplier problem. They derive a sharp structural description of the finite-part angular multiplier $M_\Omega$ associated with $T_\Omega$ .
- Key Insight: The roughness of the kernel $\Omega$ is transferred to the variation of the one-dimensional angular profile $M_\Omega$ .
- Characterization: They prove that $M_\Omega$ has bounded variation (BV) if and only if the antipodal even part of $\Omega$ , denoted $\Omega_e$ , belongs to the Hardy space $H^1(S^1)$ . This relies on the relationship between the derivative of the logarithmic part of the multiplier and the periodic Hilbert transform of $\Omega_e$ .
- Operator Bounds: By combining this BV characterization with the uniform bilinear Hilbert transform estimates of Uraltsev and Warchalski, they establish boundedness without requiring absolute continuity of the angular profile.
Directional Approach and Wavelet Analysis:
For the second endpoint, the authors investigate a critical directional condition where the power gain available for $a > 1/2$ in previous works disappears.
- Kernel Class: They introduce the class $K_{1/2, \beta}(S^1)$ , defined by a weighted integral condition with a logarithmic correction at the critical power $a=1/2$ :
  $\sup_{\xi \in S^1} \int_{S^1} \frac{|\Omega(\theta)|}{|\theta \cdot \xi|^{1/2}} \left( \log \frac{1}{|\theta \cdot \xi|} \right)^\beta d\sigma(\theta) < \infty.$
- Decomposition: The proof utilizes a dyadic decomposition of the operator and a product wavelet framework.
- Estimates: They establish endpoint Fourier decay estimates for the high-frequency pieces, showing a decay rate of $2^{-j/2} j^{-\beta}$ . This decay is converted into operator norm estimates via wavelet coefficient bounds and interpolation with polynomial bounds derived from shifted maximal and square-function arguments.

Key Contributions and Results

Structural Characterization of the Multiplier (Theorem 1.1):
The paper provides an exact characterization: for $\Omega \in L^1(S^1)$ with mean zero, the finite-part angular multiplier $M_\Omega$ is of bounded variation if and only if the antipodal even part $\Omega_e$ belongs to $H^1(S^1)$ . Specifically, $\text{Var}(M_\Omega) \simeq \|\Omega_e\|_{H^1} + \|\Omega_o\|_{L^1}$ .
The $L \log L$ Endpoint (Theorems 1.2, 1.3, 1.4):
- The authors prove that if $\Omega \in L \log L(S^1)$ (i.e., $L(\log L)^1$ ), then $T_\Omega$ is bounded on the full finite-exponent range.
- They demonstrate that the logarithmic exponent $A=1$ is optimal within the scale $L(\log L)^A$ . If $A < 1$ , boundedness fails. This is shown by constructing specific angular atoms where the operator norm grows linearly with the logarithmic scale parameter, while the kernel norm grows sub-linearly.
The Directional Endpoint (Theorem 1.6):
- Boundedness is established for kernels in the class $K_{1/2, \beta}(S^1)$ provided $\beta > \frac{3}{2} \max\{p_1, p_1', p_2, p_2'\} - 1$ .
- This result extends the boundedness mechanism of previous works (which required $a > 1/2$ ) to the critical index $a=1/2$ by inserting a logarithmic factor.
Incomparability of Kernel Classes (Theorem 1.7):
The paper proves that the Orlicz class $L(\log L)^\alpha(S^1)$ and the directional class $K_{1/2, \beta}(S^1)$ are incomparable for any $\alpha, \beta > 0$ .
- $L(\log L)^\alpha \not\subset K_{1/2, \beta}$ : Constructed by concentrating mass near a direction where the directional weight is singular but the Orlicz integral remains finite.
- $K_{1/2, \beta} \not\subset L(\log L)^\alpha$ : Constructed by distributing many small spikes uniformly around the circle, keeping directional integrals bounded while causing the Orlicz modular to diverge.

Significance and Claims
The paper claims to identify the exact rotational regularity needed for the one-dimensional bilinear problem. By linking the boundedness of the operator to the $H^1$ regularity of the even part of the kernel, it provides a precise structural criterion that allows for the use of uniform bilinear Hilbert transform estimates.

The authors assert that the $L \log L$ result is sharp, closing the gap left by previous $L^q$ ( $q>1$ ) and higher-order Orlicz results. Furthermore, the introduction of the $K_{1/2, \beta}$ class and the proof of its incomparability with Orlicz spaces highlight that different types of singular behavior (global size vs. directional concentration) require distinct endpoint criteria. The work does not claim to resolve the $L^1$ endpoint or the case $p_1=p_2=2$ (where $p=1$ ), noting that the boundary case $p=1$ lies outside the scope of the established theorems, though it records the limiting behavior of the $\beta$ threshold.

The methodology is described as novel in its separation of the even and odd parts of the kernel for the rotational approach and its use of product wavelets to handle the critical directional decay, avoiding the need for the power gain available in non-critical regimes.

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