Reduction of bilinear weighted inequalities with integration operators on the cone of nondecaying functions

Let germ{M} be the set of all measurable functions on Bbb{R}_{+}= [0,+infty). Denote by germ{M}^{+} subset germ{M} the set of nonnegative functions and by germ{M}^{+}_{i} and germ{M}^{+}_{d} the subsets of all increasing and decreasing functions respectively. Let H_{w}f(x) = int_{[0,x]} f(t) w(t) dt, where win germ{M}^{+} and fw is Lebesgue integrable on the set [0,x] for any x>0. The authors study bilinear inequalities of the following form: | H_{u_{1}}f cdot H_{u_{2}}g|_{L^{q}_{w}} leq C | f |_{L^{p_{1}}_{v_{1}}} | g|_{L^{p_{2}}_{v_{2}}},quad f,g in germ{M}^{+}_{i}, where 0< p_{1},p_{1},q leq infty and all weight functions belong to germ{M}^{+}.