Convolution estimates for measures on some complex curves

TL;DR

This paper establishes nearly optimal Lorentz space estimates for convolution operators supported on complex curves, extending Fourier restriction results and using a modified combinatorial band structure argument.

Contribution

It introduces a novel modification of the band structure argument to obtain sharp Lorentz space bounds for measures on complex curves, advancing the understanding of Fourier restriction phenomena.

Findings

Proved nearly optimal Lorentz space estimates for complex curve measures

Achieved the optimal strong type estimates as special cases

Extended Fourier restriction results to complex curves

Abstract

We consider the convolution operator for a measure supported on complex curves. The measure which we consider here is an analogue of the affine arclength measure for real curves. By modifying a combinatorial argument called the band structure argument, we prove the (nearly) optimal Lorentz space estimates. This includes the optimal strong type estimates as special cases. The complex curves we consider here are the ones considered for the Fourier restriction estimates for complex curves in \cite{BH}.