The bilinear cone multiplier on $\mathbb{R}^2\times \mathbb{R}^2$

TL;DR

This paper proves boundedness properties of a bilinear cone multiplier operator in two dimensions using geometric and maximal function techniques, extending understanding of such operators in harmonic analysis.

Contribution

It introduces a new approach to establish $L^p$ bounds for the bilinear cone multiplier via decomposition into square functions and geometric estimates.

Findings

Established $L^{p_1} imes L^{p_2} o L^{p}$ boundedness for a regularized bilinear cone multiplier.

Derived sharp $L^4$ bounds using geometric methods from Córdova and Carbery.

Connected square function bounds with maximal function estimates to achieve the main results.

Abstract

In this paper, we study the bilinear cone multiplier operator in two dimensions. We establish $L^{p_1}\times L^{p_2}\to L^{p}$ boundedness for a regularized version of this operator over a broad range of exponents satisfying the H\"older scaling condition. Our approach is based on a decomposition of the bilinear operator into square functions associated with linear cone multipliers and their variants. We derive pointwise bounds for these square functions via suitable strong maximal function estimates, and obtain sharp $L^4$ bounds using geometric methods originating in the work of C\'ordoba and Carbery. The combination of these estimates yields the $L^p$ boundedness for the bilinear cone multiplier.