Two weight inequalities for bilinear forms

Let $1\le p_0<p,q <q_0\le \infty$. Given a pair of weights $(w,\sigma)$ and a sparse family $\mathcal S$, we study the two weight inequality for the following bi-sublinear form $$B(f, g)= \sum_{Q\in\mathcal S}\langle |f|^{p_0}\rangle_Q^{\frac 1{p_0}} \langle|g|^{q_0'}\rangle_Q^{\frac 1{q_0'}}\lambda_Q\le \mathcal N\|f\|_{L^{p}(w)}\|g\|_{L^{q'}(\sigma)}.$$ When $\lambda_Q=|Q|$ and $p=q$, Bernicot, Frey and Petermichl showed that $B(f,g)$ dominates $\langle Tf, g\rangle$ for a large class of singular non-kernel operators. We give a characterization for the above inequality and then obtain the mixed $A_p$-$A_\infty$ estimates and the corresponding entropy bounds when $\lambda_Q=|Q|$ and $p=q$. We also proposed a new conjecture which implies both the one supremum conjecture and the separated bump conjecture.