Two weight inequalities for bilinear forms

Let 1 ≤ p 0 < p , q < q 0 ≤ ∞ . Given a pair of weights ( w , σ ) and a sparse family S , we study the two weight inequality for the following bi-sublinear form B ( f , g ) = ∑ Q ∈ S ⟨ | f | p 0 ⟩ Q 1 p 0 ⟨ | g | q 0 ′ ⟩ Q 1 q 0 ′ λ Q ≤ N ‖ f ‖ L p ( w ) ‖ g ‖ L q ′ ( σ ) . When λ Q = | Q | and p = q , Bernicot, Frey and Petermichl showed that B ( f , g ) dominates ⟨ T f , 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 ∞ estimates and the corresponding entropy bounds when λ Q = | Q | and p = q . We also propose a new conjecture which implies both the one supremum conjecture and the separated bump conjecture.