Improved bound for the bilinear Bochner-Riesz operator

We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}^\alpha$, $\alpha>0$ in $\mathbb{R}^d,$ $d\ge2$, which is defined by $${\mathcal B}^{\alpha}(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2\pi i x\cdot(\xi+\eta)} (1-|\xi|^2-|\eta|^2 )^{\alpha}_+ ~ \widehat{f}(\xi)\,\widehat{g}(\eta)\,d\xi d\eta.$$ 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.