  We study the weighted norm inequality of $(1,q)$-type,
  \[ \Vert \mathbf{G}\nu \Vert_{L^q(\Omega, d\sigma)} \le C \Vert \nu \Vert,
\quad \text{ for all } \nu \in \mathcal{M}^+(\Omega), \] along with its
weak-type analogue, for $0 < q < 1$, where $\mathbf{G}$ is an integral operator
associated with the nonnegative kernel $G(x,y)$. Here $\mathcal{M}^+(\Omega)$
denotes the class of positive Radon measures in $\Omega$; $\sigma, \nu \in
\mathcal{M}^+(\Omega)$, and $||\nu||=\nu(\Omega)$.
  For both weak-type and strong-type inequalities, we provide conditions which
characterize the measures $\sigma$ for which such an embedding holds. The
strong-type $(1,q)$-inequality for $0<q<1$ is closely connected with existence
of a positive function $u$ such that $u \ge \mathbf{G}(u^q \sigma)$, i.e., a
supersolution to the integral equation
  \[ u - \mathbf{G}(u^q \sigma) = 0, \quad u \in L^q_{\rm loc} (\Omega,
\sigma). \] This study is motivated by solving sublinear equations involving
the fractional Laplacian,
  \[ (-\Delta)^{\frac{\alpha}{2}} u - u^q \sigma = 0\] in domains $\Omega
\subseteq \mathbf{R}^n$ which have a positive Green function $G$, for $0 <
\alpha < n$.
