## 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.

### MSC:

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

### Keywords:

Hardy inequalities; reverse Hölder inequalities; weights

Zbl 0711.42027
Full Text:

### References:

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