Coarea Reduction, Sparse Transfer, and Geometric Recomposition for Synchronized Singular Forms

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 an architecture 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 weighted pushforward measures in the level variable. Under absolute-continuity hypotheses, this reduction admits an effective realization in the Lebesgue layer, where control of the pushforward densities yields an abstract operator criterion for feeding estimates obtained in the reduced model back into the original problem. As a first complete realization of this scheme, we transfer to the synchronized setting a one-dimensional sparse domination principle for singular truncations with Dini-smooth kernels. The final geometric recomposition then separates two regimes: a uniform regime, where global consequences follow from quantitative control of the pushforward densities, and a critical regime, where degeneration of the phases near critical values forces a localized output weighted by pullbacks.