Improved bound for the bilinear Bochner-Riesz operator

Abstract

We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator ${\mathcal{B}}^{\alpha }$, $\alpha >0$ in ${\mathbb{R}}^d,$ $d\ge 2$, which is defined by \[ {\mathcal B}^α(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2πi x\cdot(ξ+η)} (1-|ξ|^2-|η|^2 )^α_+ ~ \widehat{f}(ξ)\,\widehat{g}(η)\,dξdη.\] We make use of a decomposition which relates the estimates for ${\mathcal{B}}^{\alpha }$ to those of the square function estimates for the classical Bochner-Riesz operators. In consequence, we significantly improve the previously known bounds.

24 pages, 2 figures