Geodesic
Weighted $L_{Φ}$ integral inequalities for operators of Hardy type
Studia Mathematica, Tome 110 (1994) no. 1, pp. 35-52

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Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for $Φ_2^{-1} (ʃΦ_2(w(x)|Tf(x)|)t(x)dx) ≤ Φ_{1}^{-1}(ʃΦ_{1}(Cu(x)|f(x)|)v(x)dx)$ to hold when $Φ_1$ and $Φ_2$ are N-functions with $Φ_2∘Φ_{1}^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored. 
Steven Bloom. Weighted $L_{Φ}$ integral inequalities for operators of Hardy type. Studia Mathematica, Tome 110 (1994) no. 1, pp. 35-52. doi: 10.4064/sm-110-1-35-52
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