On two-weight norm inequalities for positive dyadic operators

Let $\sigma$ and $\omega$ be locally finite Borel measures on $\mathbb{R}^d$, and let $p\in(1,\infty)$ and $q\in(0,\infty)$. We study the two-weight norm inequality $$ \lVert T(f\sigma) \rVert_{L^q(\omega)}\leq C \lVert f \rVert_{L^p(\sigma)}, \quad \text{for all} \, \, f \in L^p(\sigma), $$ for both the positive summation operators $T=T_\lambda(\cdot \sigma)$ and positive maximal operators $T=M_\lambda(\cdot \sigma)$. Here, for a family $\{\lambda_Q\}$ of non-negative reals indexed by the dyadic cubes $Q$, these operators are defined by $$ T_\lambda(f\sigma):=\sum_Q \lambda_Q \langle f\rangle^\sigma_Q 1_Q \quad\text{ and } \quad M_\lambda(f\sigma):=\sup_Q \lambda_Q \langle f\rangle^\sigma_Q 1_Q, $$ where $\langle f\rangle^\sigma_Q:=\frac{1}{\sigma(Q)} \int_Q |f| d \sigma.$ We obtain new characterizations of the two-weight norm inequalities in the following cases: 1. For $T=T_\lambda(\cdot\sigma)$ in the subrange $qsigma$ satisfies the $A_\infty$ condition with respect to $\omega$, we characterize the inequality in terms of a simple integral condition. The proof is based on characterizing the multipliers between certain classes of Carleson measures. 2. For $T=M_\lambda(\cdot \sigma)$ in the subrange $qsummation operators $T=T_\lambda(\cdot\sigma)$ in the subrange $1summation operators by means of related inequalities for maximal operators $T=M_\lambda(\cdot \sigma)$. This maximal-type characterization is an alternative to the known potential-type characterization.