  We consider local "complementary" generalized Morrey spaces ${\dual \cal
M}_{\{x_0\}}^{p(\cdot),\om}(\Om)$ in which the $p$-means of function are
controlled over $\Om\backslash B(x_0,r)$ instead of $B(x_0,r)$, where $\Om
\subset \Rn$ is a bounded open set, $p(x)$ is a variable exponent, and no
monotonicity type conditio is imposed onto the function $\om(r)$ defining the
"complementary" Morrey-type norm. In the case where $\om$ is a power function,
we reveal the relation of these spaces to weighted Lebesgue spaces. In the
general case we prove the boundedness of the Hardy-Littlewood maximal operator
and Calderon-Zygmund singular operators with standard kernel, in such spaces.
We also prove a Sobolev type ${\dual \cal M}_{\{x_0\}}^{p(\cdot),\om}
(\Om)\rightarrow {\dual \cal M}_{\{x_0\}}^{q(\cdot),\om} (\Om)$-theorem for the
potential operators $I^{\al(\cdot)},$ also of variable order. In all the cases
the conditions for the boundedness are given it terms of Zygmund-type integral
inequalities on $\om(r)$, which do not assume any assumption on monotonicity of
$\om(r)$.
