A family of equivalent norms for Lebesgue spaces

If ψ:[0,ℓ]→[0,∞[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi :[0,\ell ]\rightarrow [0,\infty [$$\end{document} is absolutely continuous, nondecreasing, and such that ψ(ℓ)>ψ(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (\ell )>\psi (0)$$\end{document}, ψ(t)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (t)>0$$\end{document} for t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}, then for f∈L1(0,ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ > rsim $$\end{document} depends on ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} and ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}. Here by f∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^*$$\end{document} we denote the decreasing rearrangement of f. When applied with f replaced by |f|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|f|^p$$\end{document}, 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document}, there exist functions ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} so that the inequality ‖|f|p‖1,ψ,(0,ℓ)≤‖|f|p‖L1(0,ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\ell )$$\end{document}. We make an analysis on the validity of (∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(*)$$\end{document} under much weaker assumptions on the regularity of ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}, and we get a version of Hardy’s inequality which generalizes and/or improves existing results.


Introduction.
As a by-product of a characterization of an interpolation space between grand and small Lebesgue spaces, in [7, Theorems 6.2 and 6.4] (see also [1] for recent developments), it has been shown, in particular, that for 1 < p < ∞, ⎡ defined on intervals (0, ), true for 1 ≤ p < ∞, and with the constants of the equivalence independent of p. The generalization consists of the replacement of (1 − log s) −1 by a function ψ(s), and, with suitable choices of ψ, allows to get an inequality of the type f p,ψ,(0, 1) f L p (0, 1) not rougher than the classical one-dimensional integral Hardy inequality (see e.g. [11,Theorem 330], [19, (3.6) p. 23], or [20, Theorem 6 p. 726]) (1.2) in the sense that (see Proposition 3.1 for the precise statement) there exists a family of functions {ψ q } q>0 such that for every f and q sufficiently large, In spite of the fact that the main result (see We are going to prove the following then, for every f ∈ L 1 (0, ), where the constant in depends on ψ and .
Proof. At first we observe that, since ψ 0 is nondecreasing, using (2.4), we have (2.7) Since ψ is nonnegative, by (2.3), we immediately have that On the other hand, by the continuity of ψ and by (2.4), we have and therefore, setting we are allowed to fix τ ∈ (0, τ 0 ). We now estimate separately the integrals Since ψ is nondecreasing, and using the relation τ < , we have On the other hand, by (2.1), since ψ is nondecreasing and τ < τ 0 , we have ψ(τ ) < ψ( ) and therefore (2.10) Joining (2.9) and (2.10), for every fixed τ , we have and finally, setting and taking into account the identities (2.3), (2.7), we get and we are done.  [13,29]) of the left hand side of (2.5) in the assumptions of Theorem 2.1 (the equality holds by (2.2)), is an equivalent norm for L p (0, ) for 1 ≤ p < ∞. On the other hand, the left hand side of the Hardy inequality (1.2) is an equivalent norm for L p (0, 1) when 1 < p < ∞, and the degeneracy at p = 1 tells us that such an equivalent norm is not the p−convexification of a norm for L 1 (0, 1). After the proof of Theorem 2.1, and taking into account Hardy's inequality, we can assert that when 1 < p < ∞ , both the expressions We are going to show that there exist suitable choices for ψ so that the inequality f p,ψ,(0, 1) f L p (0,1) , equivalent to that one established in Theorem 2.1, is not rougher than (and therefore it cannot be deduced as a consequence of) the classical Hardy inequality.
Proof. We already observed that the right wing inequality in (3.1) holds for every f ∈ L p (0, 1), therefore we need to prove the assertion only for the left wing inequality.  .2) is not a maximum. Therefore, if we fix any function f 0 ∈ L p (0, 1), f 0 ≡ 0, we have on the other hand, by (2.7), and since ψ q (t) = (1 − log t) −q ↓ 0 a.e. as q ↑ ∞, we have 1 − ψ q (t) ↑ 1 a.e. as q ↑ ∞. By the Fatou property for Lebesgue spaces, , from which the claim follows.
The degeneracy of the Hardy inequality when p → 1 causes the appearance of a logarithm. A direct application of an integration by parts gives (see e.g. [11, 240 p. 169 It may be of interest to observe that the factor 1/t in the integrand in the left hand side is exactly ψ (t)/ψ 2 (t) if we set (3.6) Strictly speaking, this choice is not allowed in Theorem 2.1 (even considering the extension by continuity in t = 0) because in t = 1 the function ψ diverges and continuity in the closed interval [0, 1] is lost. However, we remark that the proof of Theorem 2.1 works as well changing ψ( ) into ψ( −), which is allowed to be infinite (in such case, considering, as usual, 1/ψ( −) = 0). The substitution of (3.6) into (2.5), applied with f (x) replaced by f (x) log 1 x , gives the interesting complement to (3.5) when f is nonincreasing, taking into account that the product of positive nonincreasing functions is still nonincreasing, (3.7) tells that equivalence holds in (3.5).

Remarks on the main result.
4.1. The necessity of a monotonicity assumption on ψ. In Theorem 2.1, the function ψ is assumed to be nondecreasing so that in [0, ], the existence of subintervals in which ψ is constant is allowed. However, ψ cannot be constant in the whole of [0, ] because of the assumption (2.4). Note that if ψ is a constant function, then the equivalence (2.5) cannot hold because the left hand side would be 0 (and therefore only the inequality remains true and trivial).  bounded variation (and therefore to a class of functions which may be possibly discontinuous). In fact, with the same proof of Theorem 2.1, we have

The necessity of the absolute continuity of
The validity of the integration by parts formula, in this case, holds for integrals in the Stieltjes sense (see e.g. [4, Theorem 14.10 p. 227]). In order to make the machinery work, we need, however, that ψ is positive and strictly increasing, so that also −1/ψ is positive and strictly increasing (which implies that it is of bounded variation).

Removing any regularity assumption on ψ.
If ψ is not assumed to be absolutely continuous, but just nondecreasing, the equivalence (in general) does not hold (as we already saw, considering the Cantor ternary function). However, the inequality in (2.5)

Relevant special cases.
In the case of the functions ψ q in Proposition 3.1, we have ψ q (t) = (1 − log t) −q , t ∈ (0, 1), q > 0, and therefore When q = 1, and (2.5) reduces to (1.1). Elementary but tedious computations show that when ψ q (t) is multiplied by a power t λ , λ 0, the inequality proved in Theorem 2.1 fits into the special case of b = 1 and Φ decreasing of a celebrated inequality proved in Bennett-Rudnik [2, Theorem 6.4] (see also [7,Theorem 2.1]). It is interesting to observe that, in this special situation, Theorem 2.1 guarantees the validity of the reverse inequality, too.

Comparison with some existing results.
The literature about integral inequalities for real valued functions, more or less related to variants and generalizations of the Hardy inequality, is huge: it starts even before Hardy (the references in the celebrated [11] are a proof), it grew in the sixties and in the seventies of the last century, and it continues to grow until now, see e.g. [14,15,22]. The development has been in several directions, among which the framework in Banach spaces other than Lebesgue spaces (see e.g. the treatment [19,Chapter 11], [8], and [13]). Any comparison with existing results has to be incomplete, however, we will try to mention some references at our knowledge. It will be of help to record some features of (2.5) in Theorem 2.1: