On plate decompositions of cone multipliers

TL;DR

This paper improves Wolff's inequality for cone multipliers using bilinear restriction results, leading to better estimates on square functions and a new L^4 bound for the cone multiplier in three dimensions.

Contribution

It introduces improved bounds for cone multipliers by leveraging bilinear restriction techniques, enhancing previous harmonic analysis results.

Findings

Enhanced Wolff's inequality range for cone multipliers

Improved square function estimates in low dimensions

New L^4 bound for cone multiplier in three dimensions

Abstract

An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for a variety of problems in harmonic analysis. We observe that the range in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L^4 bound for the cone multiplier operator in three dimensions.