Weighted Modular Inequalities for Hardy Type Operators

Given weight functions θ, w, ρ and v, the weighted modular inequality Q − 1 ( ∫ 0 ∞ Q ( θ ( x ) T f ( x ) ) w ( x ) d x ) ⩽ P − 1 ( ∫ 0 ∞ P ( C ρ ( x ) f ( x ) ) v ( x ) d x ) is characterized. Here Q is a strictly increasing function with Q(0) = 0, Q(∞) = ∞ and 2Q(x) ⩽ Q(C x), P is a Young's function, and T is the Hardy operator or a Hardy type operator. In particular, a characterizing condition for the Hardy type operator to map Lp(w) to Lq(v) when 0 < q < 1 ⩽ p < ∞ is deduced. In addition, a new proof for the Maz'ja-Sinnamon theorem is given, and weighted Lorentz norm inequalities for Hardy type operators are established. 1991 Mathematics Subject Classification: primary 26D15, 42B25; secondary 26A33, 46E30.