$L^p$ Regularity Estimates for a Class of Integral Operators with Fold Blowdown Singularities

TL;DR

This paper establishes sharp $L^p$ regularity estimates for certain integral operators with fold and blowdown singularities, advancing understanding of their boundedness properties in harmonic analysis.

Contribution

It introduces new $L^p$ regularity results for generalized Radon transforms with complex singularities, utilizing advanced decoupling inequalities and oscillatory integral estimates.

Findings

Proves sharp $L^p$ bounds for integral operators with fold and blowdown singularities.

Employs decoupling inequalities by Wolff and Bourgain-Demeter in the analysis.

Provides a framework for analyzing singular Radon transforms in three dimensions.

Abstract

We prove sharp $L^p$ regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated to a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain-Demeter for plate decompositions of thin neighborhoods of cones and $L^2$ estimates for related oscillatory integrals.