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

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 Hölder 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 Córdoba and Carbery. The combination of these estimates yields the $L^p$ boundedness for the bilinear cone multiplier.