Weighted Modular Inequalities for Hardy Type Operators

Given weight functions , w, and v, the weighted modular inequality is characterized. Here Qis a strictly increasing function with Q(0) = 0, Q() = and2Q(x) Q(C x), P is a Young's function, and T is the Hardy operatoror a Hardy type operator. In particular, a characterizing conditionfor the Hardy type operator to map L^p(w) to L^q(v) when 0 <q < 1 p < is deduced. In addition, a new proof for theMaz'ja-Sinnamon theorem is given, and weighted Lorentz norminequalities for Hardy type operators are established. 1991Mathematics Subject Classification: primary 26D15, 42B25; secondary26A33, 46E30.