Two weight inequalities for bilinear forms

Let $1\le p_0<p,q<q_0\le \mathrm{\infty }$. 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 $$B(f,g)=\sum _{Q\in \mathcal{S}}⟨\mathrm{\mid }f{\mathrm{\mid }}^{p_0}{⟩}_Q^{\frac{1}{p_0}}⟨\mathrm{\mid }g{\mathrm{\mid }}^{q_0^{\mathrm{\prime }}}{⟩}_Q^{\frac{1}{q_0^{\mathrm{\prime }}}}{\lambda }_Q\le \mathcal{N}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{q^{\mathrm{\prime }}}(\sigma )}\mathrm{.}$$ When ${\lambda }_Q=\mathrm{\mid }Q\mathrm{\mid }$ and $p=q$, Bernicot, Frey and Petermichl showed that $B(f,g)$ dominates $⟨Tf,g⟩$ for a large class of singular non-kernel operators. We give a characterization for the above inequality and then obtain the mixed $A_p$-$A_{\mathrm{\infty }}$ estimates and the corresponding entropy bounds when ${\lambda }_Q=\mathrm{\mid }Q\mathrm{\mid }$ and $p=q$. We also proposed a new conjecture which implies both the one supremum conjecture and the separated bump conjecture.