A family of equivalent norms for Lebesgue spaces

If ψ: [0 , ℓ] → [0 , ∞[is absolutely continuous, nondecreasing, and such that ψ(ℓ) > ψ(0) , ψ(t) > 0 for t> 0 , then for f∈ L1(0 , ℓ) , we have ‖f‖1,ψ,(0,ℓ):=∫0ℓψ′(t)ψ(t)2∫0tf∗(s)ψ(s)dsdt≈∫0ℓ|f(x)|dx=:‖f‖L1(0,ℓ),(∗)where the constant in ≳ depends on ψ and ℓ. Here by f∗ we denote the decreasing rearrangement of f. When applied with f replaced by | f| p, 1 < p< ∞, there exist functions ψ so that the inequality ‖|f|p‖1,ψ,(0,ℓ)≤‖|f|p‖L1(0,ℓ) is not rougher than the classical one-dimensional integral Hardy inequality over bounded intervals (0 , ℓ). We make an analysis on the validity of (∗) under much weaker assumptions on the regularity of ψ, and we get a version of Hardy’s inequality which generalizes and/or improves existing results.