  Let $\sigma$, $\omega$ be measures on $\mathbb{R}^d$, and let
$\{\lambda_Q\}_{Q\in\mathcal{D}}$ be a family of non-negative reals indexed by
the collection $\mathcal{D}$ of dyadic cubes in $\mathbb{R}^d$. We characterize
the two-weight norm inequality, \begin{equation*} \lVert
T_\lambda(f\sigma)\rVert_{L^q(\omega)}\le C \, \lVert f
\rVert_{L^p(\sigma)}\quad \text{for every $f\in L^p(\sigma)$,} \end{equation*}
for the positive dyadic operator \begin{equation*} T_\lambda(f\sigma):=
\sum_{Q\in \mathcal{D}} \lambda_Q \, \Big(\frac{1}{\sigma(Q)} \int_Q
f\mathrm{d}\sigma\Big) \, 1_Q \end{equation*} in the difficult range $0<q<1 \le
p<\infty$ of integrability exponents. This range of the exponents $p, q$
appeared recently in applications to nonlinear PDE, which was one of the
motivations for our study.
  Furthermore, we introduce a scale of discrete Wolff potential conditions that
depends monotonically on an integrability parameter, and prove that such
conditions are necessary (but not sufficient) for small parameters, and
sufficient (but not necessary) for large parameters.
  Our characterization applies to Riesz potentials $I_\alpha (f \sigma) =
(-\Delta)^{-\frac{\alpha}{2}} (f\sigma) $ ($0<\alpha<d$), since it is known
that they can be controlled by model dyadic operators. The weighted norm
inequality for Riesz potentials in this range of $p, q$ has been characterized
previously only in the special case where $\sigma$ is Lebesgue measure.
