Given weight functions $\theta$, $w$, $\rho$ and $v$, the weighted modular inequality $$ Q^{-1}\bigg(\int_{0}^{\infty} Q(\theta(x)Tf(x)) w(x)\, dx\bigg) \leq P^{-1}\bigg(\int_{0}^{\infty} P(C \rho(x) f(x)) v(x)\, dx\bigg)$$ is characterized. Here $Q$ is a strictly increasing function with $Q(0) = 0$, $Q(\infty) = \infty$ and $2Q(x) \leq Q(C x)$, $P$ 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 $L^{p}(w)$ to $L^{q}(v)$ when $0 < q < 1 \leq p < \infty$ 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.