  We study weighted norm inequalities of $(1,q)$- type for $0<q<1$, $\Vert
\mathbf{G} \nu \Vert_{L^q(\Omega, d \sigma)} \le C \, \Vert \nu \Vert, \quad
\text{for all positive measures $\nu$ in $\Omega$},$ along with their weak-type
counterparts, where $\Vert \nu \Vert=\nu(\Omega)$, and $G$ is an integral
operator with nonnegative kernel, $\mathbf{G} \nu(x) = \int_\Omega G(x, y) d
\nu(y).$
  These problems are motivated by sublinear elliptic equations in a domain
$\Omega\subset\mathbb{R}^n$ with non-trivial Green's function $G(x, y)$
associated with the Laplacian, fractional Laplacian, or more general elliptic
operator. We also treat fractional maximal operators $M_\alpha$ ($0\le
\alpha<n$) on $\mathbb{R}^n$, and characterize strong- and weak-type
$(1,q)$-inequalities for $M_\alpha$ and more general maximal operators, as well
as $(1,q)$-Carleson measure inequalities for Poisson integrals.
