On two-weight norm inequalities for positive dyadic operators

Let $\sigma$ and $\omega$ be locally finite Borel measures on ${\mathbb{R}}^d$, and let $p\in (1,\mathrm{\infty })$ and $q\in (0,\mathrm{\infty })$. We study the two-weight norm inequality $$\parallel T(f\sigma ){\parallel }_{L^q(\omega )}\le C\parallel f{\parallel }_{L^p(\sigma )},\text{for all}f\in L^p(\sigma ),$$ 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 $$T_{\lambda }(f\sigma ):=\sum _Q{\lambda }_Q⟨f{⟩}_Q^{\sigma }{1}_Q\text{ and }{M}_{\lambda }(f\sigma ):=\underset{Q}{\sup }{\lambda }_Q⟨f{⟩}_Q^{\sigma }{1}_Q,$$ 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 $q<p$. Under the additional assumption that $\sigma$ satisfies the $A_{\mathrm{\infty }}$ 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 $q<p$. We introduce a scale of simple conditions that depends on an integrability parameter and show that, on this scale, the sufficiency and necessity are separated only by an arbitrarily small integrability gap.
3. For the summation operators $T=T_{\lambda }(\cdot \sigma )$ in the subrange $1<q<p$. We characterize the inequality for summation 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.