  We study weighted norm inequalities of $(p,r)$-type, $ \Vert \mathbf{G} (f \,
d \sigma) \Vert_{L^r(\Omega, d\sigma)} \le C \Vert f \Vert_{L^p(\Omega,
\sigma)}, \quad \forall \, f \in L^p(\sigma),$ for $0 < r < p$ and $p>1$, where
$\mathbf{G}(f d \sigma)(x)=\int_\Omega G(x, y) f(y) d \sigma(y)$ is an integral
operator associated with a nonnegative kernel $G$ on $\Omega\times \Omega$, and
$\sigma$ is a locally finite positive measure in $\Omega$. We show that this
embedding holds if and only if $\int_\Omega (\mathbf{G}
\sigma)^{\frac{pr}{p-r}} d \sigma<+\infty,$ provided $G$ is a quasi-symmetric
kernel which satisfies the weak maximum principle.
  In the case $p=\frac{r}{q}$, where $0<q<1$, we prove that this condition
characterizes the existence of a non-trivial solution (or supersolution) $u \in
L^r(\Omega, \sigma)$, for $r>q$, to the the sublinear integral equation $ u -
\mathbf{G}(u^q \, d \sigma) = 0, \quad u \ge 0.$ We also give some
counterexamples in the end-point case $p=1$, which corresponds to solutions $u
\in L^q (\Omega, \sigma)$ of this integral equation. These problems appear in
the investigation of weak solutions to the sublinear equation involving the
(fractional) Laplacian, $(-\Delta)^{\alpha} u - \sigma \, u^q = 0, \quad u \ge
0,$ for $0<q<1$ and $0 < \alpha < \frac{n}{2}$ in domains $\Omega \subseteq
\mathbb{R}^n$ with a positive Green function.
