Two weight inequalities for bilinear forms

Abstract

Let 1 ≤ p0< p, q< q0≤ ∞. Given a pair of weights (w, σ) and a sparse family S, we study the two weight inequality for the following bi-sublinear form (Formula presented.) When λQ= | Q| 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 Ap- A∞ estimates and the corresponding entropy bounds when λQ= | Q| and p= q. We also propose a new conjecture which implies both the one supremum conjecture and the separated bump conjecture.