  In this paper we study maximal and square functions associated with bilinear
Bochner-Riesz means at the critical index. In particular, we prove that they
satisfy weighted estimates from $L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow
L^p(v_w)$ for bilinear weights $(w_1,w_2)\in A_{\vec{P}}$ where $p_1,p_2>1$ and
$\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Also, we show that both the
operators fail to satisfy weak-type estimates at the end-point
$(1,1,\frac{1}{2})$.
