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 ${\mathbb{R}}^d,$ $d\ge 2$, which is defined by $${\mathcal{B}}^{\alpha }(f,g)={\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}{e}^{2\pi ix\cdot (\xi +\eta )}(1-\mathrm{\mid }\xi {\mathrm{\mid }}^2-\mathrm{\mid }\eta {\mathrm{\mid }}^2{)}_{+}^{\alpha }\widehat{f}(\xi )\widehat{g}(\eta )d\xi d\eta \mathrm{.}$$ 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.