If ψ:[0,ℓ]→[0,∞[\documentclass[12pt]{minimal}
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\begin{document}$$\psi :[0,\ell ]\rightarrow [0,\infty [$$\end{document} is absolutely continuous, nondecreasing, and such that ψ(ℓ)>ψ(0)\documentclass[12pt]{minimal}
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\begin{document}$$\psi (\ell )>\psi (0)$$\end{document}, ψ(t)>0\documentclass[12pt]{minimal}
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\begin{document}$$\psi (t)>0$$\end{document} for t>0\documentclass[12pt]{minimal}
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\begin{document}$$t>0$$\end{document}, then for f∈L1(0,ℓ)\documentclass[12pt]{minimal}
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\begin{document}$$f\in L^1(0,\ell )$$\end{document}, 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),(∗)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert f\Vert _{1,\psi ,(0,\ell )}:=\int \limits _0^\ell \frac{\psi '(t)}{\psi (t)^2}\int \limits _0^tf^*(s)\psi (s)dsdt\approx \int \limits _0^\ell |f(x)|dx=:\Vert f\Vert _{L^1(0,\ell )},\quad (*) \end{aligned}$$\end{document}where the constant in ≳\documentclass[12pt]{minimal}
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\begin{document}$$ > rsim $$\end{document} depends on ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} and ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}. Here by f∗\documentclass[12pt]{minimal}
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\begin{document}$$f^*$$\end{document} we denote the decreasing rearrangement of f. When applied with f replaced by |f|p\documentclass[12pt]{minimal}
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\begin{document}$$|f|^p$$\end{document}, 1<p<∞\documentclass[12pt]{minimal}
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\begin{document}$$1<p<\infty $$\end{document}, there exist functions ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} so that the inequality ‖|f|p‖1,ψ,(0,ℓ)≤‖|f|p‖L1(0,ℓ)\documentclass[12pt]{minimal}
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\begin{document}$$\Vert |f|^p\Vert _{1,\psi ,(0,\ell )}\le \Vert |f|^p\Vert _{L^1(0,\ell )}$$\end{document} is not rougher than the classical one-dimensional integral Hardy inequality over bounded intervals (0,ℓ)\documentclass[12pt]{minimal}
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\begin{document}$$(0,\ell )$$\end{document}. We make an analysis on the validity of (∗)\documentclass[12pt]{minimal}
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\begin{document}$$(*)$$\end{document} under much weaker assumptions on the regularity of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}, and we get a version of Hardy’s inequality which generalizes and/or improves existing results.
