A family of equivalent norms for Lebesgue spaces
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Archiv der Mathematik
A family of equivalent norms for Lebesgue spaces Alberto Fiorenza and Pankaj Jain Abstract. If ψ : [0, ] → [0, ∞[ is absolutely continuous, nondecreasing, and such that ψ() > ψ(0), ψ(t) > 0 for t > 0, then for f ∈ L1 (0, ), we have t ψ (t) ∗ f (s)ψ(s)dsdt ≈ |f (x)|dx =: f L1 (0,) , (∗) f 1,ψ,(0,) := ψ(t)2 0
0
0
where the constant in depends on ψ and . 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,) ≤ |f |p L1 (0,) is not rougher than the classical one-dimensional integral Hardy inequality over bounded intervals (0, ). 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. Mathematics Subject Classification. 26D10, 26D15, 46E30. Keywords. Integral inequalities, Lebesgue spaces, Weighted Lebesgue spaces, Banach function space norms, GΓ spaces, Absolutely continuous functions, Monotone functions.
1. 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 < ∞, ⎞ ⎤ p1 ⎛ 1 ⎞ p1 ⎡ 1⎛ t ⎣ ⎝ (1 − log s)−1 f ∗ (s)p ds⎠ dt ⎦ ≈ ⎝ |f (x)|p dx⎠ , (1.1) t 0
0
0
where f ∗ denotes the decreasing rearrangement of f . The goal of this note is to give a direct proof of a generalized version of this equivalence for functions
Arch. Math.
A. Fiorenza and P. Jain
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 Lp (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]) ⎛ x ⎞p p 1 1 p 1 ⎝ |f (t)|dt⎠ dx |f (x)|p dx, (1.2) x p−1 0
0
0
in the sense that (see Proposition 3.1 for the precise statement) there exists a family of functions {ψq }q>0 such that for every f and q sufficiently large, ⎛ x ⎞p p 1 1 p−1 1 p ⎝ f (t)dt⎠ dx f p,ψq ,(0,1) f (x)p dx. (1.3) p x 0
0
0
In spite of the fact that the main result (see Theorem 2.1) consists of a kind of inequality quite familiar in the literature (we quote, for instance, [21, Example 6.17 (ii) p. 330]; see also the final section), we believe that it can inspire new equivalent norms for several concrete examples of Banach function spaces, whose norm is built from that one of the Lebesgue spaces. 2. The main result. Let 0 < < ∞. Denote by M the set of the a.e. finite, Lebesgue measurable functions defined in (0, ). For every function f ∈ M, its distribution function λf : (0, ∞) → R+ is defined by λf (s) := |{x ∈ (0, ) : |f (x)| > s}|,
s > 0,
where by
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