$L^2$-estimates for singular oscillatory integral operators

TL;DR

This paper establishes $L^2$ bounds for singular oscillatory integral operators over hypersurfaces, including maximal functions, with applications to the Helmholtz eigenvalue problem in three dimensions.

Contribution

It provides new $L^2$ estimates for singular oscillatory integrals with linear and nonlinear phases over oscillating hypersurfaces, extending previous results.

Findings

Proved $L^2$ bounds for oscillatory integrals over hypersurfaces.

Established estimates for the associated maximal functions.

Applied results to the Helmholtz eigenvalue problem in $R^3$.

Abstract

In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of $L^2 \mapsto L^2$ type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in $\mathbb{R}^3$.