An alternate approach to bilinear rough singular integrals

TL;DR

This paper introduces a novel method using local Fourier series expansion to establish $L^p$-boundedness of bilinear rough singular integrals, offering a new proof and sharper estimates compared to traditional wavelet-based techniques.

Contribution

It presents a new approach based on local Fourier series expansion, providing a self-contained proof and sharp estimates for bilinear rough singular integrals, diverging from wavelet decomposition methods.

Findings

Proved $L^p$-boundedness in dimension one for optimal exponents.

Established sharp $L^p$-estimates for maximally truncated integrals.

Introduced a new method that simplifies and extends previous results.

Abstract

The goal of this paper is to provide a new approach to address the $L^p-$boundedness of bilinear rough singular integral operators. This approach relies on local Fourier series expansion of input functions leading to trilinear estimates with desired decay in the frequency parameter. This approach departs from the existing methods of the wavelet decomposition of the multiplier employed in the work of Grafakos, He and Honz\'ik and in a series of subsequent papers in the context of bilinear rough singular integrals. With this new approach, we provide a new and self contained proof of $L^p-$boundedness of bilinear rough singular integral operators in dimension one for the optimal range of exponents. Furthermore, this approach allows us to prove sharp $L^p-$estimates for maximally truncated bilinear rough singular integrals when the kernel is supported away from the diagonal in the plane.