Sublinear equations and Schur's test for integral operators

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.