  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.
