  A local median decomposition is used to prove that a weighted local mean of a
function is controlled by a weighted local mean of its local sharp maximal
function. Together with (a local version of) the estimate
$M^{\sharp}_{0,s}(Tf)(x) \le c\,Mf(x)$ for Calder\'{o}n-Zygmund singular
integral operators, this allows us to express the local weighted integral
control of $Tf$ by $Mf$. Similar estimates hold for $T$ replaced by singular
integrals with kernels satisfying H\"{o}rmander-type conditions or integral
operators with homogeneous kernels, and $M$ replaced by an appropriate maximal
function $M_T$. Using sharper bounds in the local median decomposition we prove
two-weight, $L^p_v$-$L^q_w$ estimates for singular integral operators for
$1<p\le q<\infty$. In all cases, the results include weights that are not
necessarily $A_{\infty}$. The local nature of these estimates leads to results
involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.
