Coarea reduction, transfer, and geometric recomposition for synchronized singular forms

TL;DR

This paper develops a framework for analyzing synchronized bilinear forms with singular kernels, enabling reduction, transfer, and geometric recomposition to improve understanding and control of such forms in harmonic analysis.

Contribution

It introduces an exact reduction and transfer scheme for synchronized kernels, providing a new approach to analyze their singular and geometric components.

Findings

Established an exact reduction at the measure level.

Transferred sparse domination results to the synchronized setting.

Separated regimes for uniform and localized geometric analysis.

Abstract

We study truncated bilinear forms associated with synchronized kernels \[ K(x,y)=k(\phi(x),\psi(y)), \] where the singularity is governed by a one-dimensional kernel $k$, while the geometry is encoded by the phases $\phi$ and $\psi$. The central result of the paper is a framework of exact reduction, analytic transfer, and geometric recomposition for this class of forms. First, we obtain an exact reduction at the level of pushforward measures and data-weighted pushforward measures in the level variable. Under absolute continuity hypotheses, this reduction admits a realization on the Lebesgue layer, where control of the pushforward densities yields an abstract operatorial criterion for reinjecting into the original problem estimates obtained for the reduced model. As a first complete realization of this scheme, we transfer to the synchronized setting a one-dimensional sparse domination for singular truncations with Dini-smooth kernels. The final geometric recomposition then separates two regimes: a uniform regime, in which global consequences are obtained under quantitative control of the pushforward densities, and a critical regime, in which the degeneration of the phases near the critical values forces a localized and pullback-weighted output.