A Hardy inequality and applications to reverse Hölder inequalities for weights on \(\mathbb{R}\). (English) Zbl 1396.26035

The following Hardy-type inequality is obtained:
Theorem. Let \((a_n)_n\) be a sequence of non-negative real numbers. We define for every sequence \((\lambda_n)_n\) of positive numbers the following quantities \(A_n=\lambda_1a_1+\dots +\lambda_na_n\) and \(\Lambda_n=\lambda_1 + \dots +\lambda_n\). Then the following inequality is true: \[ \sum_{n=1}^N \lambda_n \left( \frac{A_n}{\Lambda_n} \right)^p \leq \frac{p}{p-1} \sum_{n=1}^N \lambda_n a_n \left( \frac{A_n}{\Lambda_n} \right)^{p-1} -\frac{1}{p-1} \Lambda_n \left(\frac{A_N}{\Lambda_N}\right)^p, \] for any \(N\in \mathbb{N}\).
As a consequence of the above-mentioned theorem, we have the following result.
Corollary. Let \(g:[0,1] \rightarrow \mathbb{R}^+\) be integrable function, \(p>1\) and additionally assume that \(\int_0^1g=f\). Then the following inequality is true \[ \int_0^1 \left( \frac 1t \int_0^t g \right)^p dt \leq \frac{p}{p-1} \int_0^1 \left(\frac 1t \int_0^t g\right)^{p-1}g(t)dt -\frac{1}{p-1}f^p. \]
Also, the following general result is obtained.
Theorem. Let \(g:[0,1] \rightarrow \mathbb{R}^+\) be an integrable function, \(p>1\) and additionally assume that \(\int_0^1g=f\). Then, the following inequality is true, for any \(q\) such that \(1\leq q\leq p\) \[ \int_0^1 \left( \frac 1t \int_0^t g \right)^p dt \leq \left(\frac{p}{p-1}\right)^q \int_0^1 \left(\frac 1t \int_0^t g\right)^{p-q}g^q(t)dt -\frac{q}{p-1}f^p. \] Moreover, the inequality is sharp in the sense that, the constant \((p/(p-1))^q\) cannot be decreased, while the constant \(q/(p-1)\) cannot be increased for any fixed \(f\).
As an application, the author obtained the exact best possible range of \(p>q\) such that any non-increasing \(g\) which satisfies a reverse Hölder inequality with exponent \(q\) and constant \(c\) upon the subintervals of \((0,1]\) should additionally satisfy a reverse Hölder inequality with exponent \(p\) and in general different constant \(c'\). Namely, the following result is given.
Theorem. Let \(g:(0,1] \rightarrow \mathbb{R}^+\) be non-increasing satisfying a reverse Hölder inequality with exponent \(q>1\) and constant \(c \geq 1\) for all intervals of the form \((0,t]\), i.e., \[ \frac 1t \int_0^t g^q \leq c\cdot \left( \frac 1t \int_0^t g\right)^q \] holds for any \(t\in (0,1]\). Then, for every \(p\in [q,p_0)\) the inequality \[ \frac 1t \int_0^t g^p \leq c'\cdot \left( \frac 1t \int_0^t g\right)^p \] is true for any \(t\in (0,1]\), where \(c'=c'(p,q,c)\) and \(p_0>q\) is a root of the following equation \[ \frac{p_0-q}{p_0}\cdot \left( \frac{p_0}{p_0-1}\right)^q \cdot c=1. \] As a consequence \(g\in L^p\) for every \(p\in [q,p_0)\). Moreover, the result is sharp, i.e., the value of \(p_0\) cannot be decreased.
The above-mentioned result about the reverse Hölder inequalities are already known, see [L. D’Apuzzo and C. Sbordone, Rend. Mat. Appl., VII. Ser. 10, No. 2, 357–366 (1990; Zbl 0711.42027)], but here an alternative proof is given based on the obtained Hardy-type inequalities.


26D15 Inequalities for sums, series and integrals
42B25 Maximal functions, Littlewood-Paley theory


Zbl 0711.42027
Full Text: DOI arXiv


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