Weighted weak type $(1,1)$ estimates for oscillatory singular integrals with Dini kernels

Abstract: We consider $A_1$-weights and prove weighted weak type $(1,1)$ estimates for oscillatory singular integrals with kernels satisfying a Dini condition.

• The paper derives sharp weighted weak type (1,1) estimates for oscillatory singular integrals with Dini-type regularity, improving classical results.
• It employs a Calderón-Zygmund decomposition and polynomial phase induction to control rough kernel oscillations and endpoint bounds.
• The study extends endpoint bounds to Muckenhoupt A1 weighted spaces, offering robust constants dependent on polynomial degree and kernel regularity.

Weighted Weak Type $(1,1)$ Estimates for Oscillatory Singular Integrals with Dini Kernels

Introduction and Motivation

This paper rigorously investigates the weighted weak type $(1,1)$ bounds for oscillatory singular integral operators with kernels subject to Dini-type regularity, focusing on the endpoint theory for such operators. The operators considered are of the form

$$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$

where $K$ satisfies size and smoothness estimates (of Calderón-Zygmund type), and $P(x, y)$ is a real polynomial. The central focus is on the scenario where the kernel $K$ may be "rough," i.e., may lack classical smoothness properties, yet satisfies a Dini-type condition—an assumption weaker than Hölder continuity. Moreover, the study is performed in the setting of weighted spaces with Muckenhoupt $A_1$ weights, which is central for extrapolation theory and applications to harmonic analysis and PDEs.

Prior work by Ricci-Stein and Chanillo-Christ established strong-type and weak-type estimates (respectively) for oscillatory singular integrals in the unweighted setting, primarily with classical smooth kernels. Extensions to the weighted case or with rough kernels required substantially more delicate harmonic analytic machinery, particularly in controlling kernel oscillations and singularities simultaneously. The current work sharpens existing results both by weakening the kernel regularity requirement (from classical conditions to Dini-type) and by refining the weighted endpoint estimate to be essentially sharp with respect to the natural parameters.

Main Results

The paper establishes two principal theorems:

1. Weighted Weak Type (1,1) Bound for Oscillatory Singular Integrals with Dini Kernels ($D_r$-condition):

Let $K$ satisfy the $D_r$-regularity condition (for some $(1,1)$0), and let $(1,1)$1, $(1,1)$2. Then, with $(1,1)$3 as described, there exists a constant $(1,1)$4 (depending only on the degree of $(1,1)$5, $(1,1)$6, $(1,1)$7, $(1,1)$8, $(1,1)$9, and $$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$0) such that

$$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$1

This improves on results where either the kernel is smoother or the weighted space is limited to unweighted ($$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$2) or stronger $$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$3 classes.

2. Unweighted Endpoint Estimate under Minimal Dini Regularity ($$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$4-condition):

For $$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$5 satisfying the $$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$6-condition,

$$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$7

Key Contrasts: These theorems relax the kernel's regularity from classical smoothness to a Dini-type condition and permit endpoint weak-type bounds with respect to Muckenhoupt weights depending on the integrability exponent associated with the kernel. The results are uniform over polynomial phases and weight parameters.

Technical Approach

Kernel Dini-type Regularity

The paper introduces and utilizes the $$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$8-condition (and its limiting case $$T(f)(x) = \mathrm{p.v.} \int_{\mathbb{R}^n} e^{iP(x, y)} K(x - y) f(y) \, dy,$$9), which controls the oscillation and averaging of the kernel in $K$0 over annuli, in the spirit of the classical Dini condition for continuity,

$K$1

where $K$2 is a local modulus of continuity for $K$3 averaged in $K$4 over scales.

Calderón-Zygmund Decomposition and Polynomial Phase Induction

The proof architecture heavily leverages the Calderón-Zygmund decomposition adapted to the weighted setting, splitting the input function into "good" and "bad" components. The oscillation from the polynomial phase $K$5 is handled using fine-scale induction on the bidegree $K$6 of the polynomial $K$7, following the method pioneered by Ricci-Stein and Chanillo-Christ. This induction exploits the algebraic structure of polynomials, spectral localization, and multimodal decompositions.

Treatment of Oscillation and Locality

To control the nonlocal oscillatory contributions, the kernel is decomposed into local ($K$8) and nonlocal ($K$9) parts, with a fine analysis of associated operators. The case analysis differentiates between contributions where the oscillation from $P(x, y)$0 can be absorbed by cancelling principal values and the residual terms that require careful geometric measure theoretic estimates.

Weighted Interpolation and Endpoint Analysis

A significant innovation is the application of Vargas's weighted interpolation with a change-of-measure argument in the context of rough operators. This technique bridges the gap between known $P(x, y)$1 bounds and the desired endpoint $P(x, y)$2 estimates, a procedure that is fundamental due to the failure of strong-type bounds at the endpoint.

Geometric Lemmas for Polynomial Phases

The analysis of polynomial phase sets is underpinned by geometric measure theory: the paper proves a technical lemma estimating the measure of tubular neighborhoods around polynomial zero sets, based on dyadic cube decompositions and the structure of real algebraic varieties. This becomes central for handling the oscillatory integrations arising from the rough singularities, tying together real-variable methods and the geometry of polynomials.

Implications and Significance

The results provide near-optimal endpoint weighted estimates in the weak-type regime for oscillatory singular integrals in the presence of roughness, extending the classical Calderón-Zygmund- and Stein-type singular integral theory to a broader class of kernels. The extension to the weighted setting with $P(x, y)$3 weights and Dini-type kernels is nontrivial; these spaces are central to many applications (e.g., boundary value problems for PDEs, weighted inequalities in signal processing, and extrapolation theory). The double induction on the polynomial's bidegree and the careful localization of oscillatory behavior represent a consolidation of algebraic and analytic techniques that may inform future analysis of variable coefficient singular integrals, polynomial Carleson operators, and multi-parameter oscillatory integrals.

Notably, the dependence of constants only on explicit quantifiable parameters (degree of $P(x, y)$4, $P(x, y)$5, $P(x, y)$6, $P(x, y)$7 characteristic) renders the results robust for applications where such data are structural, such as in quantitative harmonic analysis or in the study of operators invariant under group actions.

Possible Directions for Future Research

• Extension to multilinear oscillatory singular integrals under analogous Dini conditions,
• Further exploration of weighted strong-type bounds below the natural endpoint and possible self-improvement phenomena for operator classes under weaker kernel regularity,
• Application of the developed geometric measure results to oscillatory Radon-type transforms and other operators with nontrivial phase interactions,
• Quantitative estimates of the constants involved for fine extrapolation theory and applications in PDEs sensitive to weight parameters.

Conclusion

The paper advances the understanding of weighted endpoint bounds for oscillatory singular integrals with rough, Dini-type kernels. It develops a robust inductive framework for controlling such operators in the presence of oscillatory polynomial phases and weights, substantially generalizing and sharpening earlier results. The combination of analytic, geometric, and algebraic tools deployed here signals a mature direction for further work at the interface of classical harmonic analysis, real algebraic geometry, and weighted inequalities.

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Reference:

"Weighted weak type $P(x, y)$8 estimates for oscillatory singular integrals with Dini kernels" (2604.25218)