The reverse Holder inequality for an elementary function

  • A.O. Korenovskii Odesa I. I. Mechnikov National University
Keywords: Reverse H ̈older inequality; Step function; Extreme parameters

Abstract

For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\in\mathbb R$ is defined by

\smallskip\centerline{$\displaystyle
\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),
\qquad
\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$}

Assume that $0<A<B$, $0<\theta<1$ and consider the step function
$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$, where $\chi_E$ is the characteristic function of the set $E$.

Let $-\infty<p<q<+\infty$. The main result of this work consists in finding the term

\smallskip\centerline{$\displaystyle
C_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$}

\smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\theta_{p<q,A<B}$ with respect to $\beta=B/A\in(1,+\infty)$.
The cases $p=0$ or $q=0$ are considered separately.

The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].

Author Biography

A.O. Korenovskii, Odesa I. I. Mechnikov National University

Faculty of Mathematics, Physics and Information Technologies, Odesa I. I. Mechnikov National University

References

B. Muckenhoupt, Weighted inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 533–565.

F.W. Gehring, The Lp-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265–273.

L.G. Gurov, Yu.G. Reshetnyak, An analogue of the concept of functions with bounded mean oscillation, Siberian Math. J., 17 (1976), №3, 417—422.

A. Korenovskyi, The Gurov—Reshetnyak inequality on semi-axes, Ann. Mat. Pura Appl., 195 (2016), №2, 659–680.

A. Korenovskii, Mean oscillations and equimeasurable rearrangements of functions, Lecture Notes of Unione Mat. Ital., V.4, Springer, Berlin, 2007. – 189 p.

A.A. Korenovskyi, Estimation of the rate of decrease (vanishing) of a function in terms of relative oscillations, Ukr. Mat. J., 71 (2019), №2, 278–295.

Published
2021-10-23
How to Cite
1.
Korenovskii A. The reverse Holder inequality for an elementary function. Mat. Stud. [Internet]. 2021Oct.23 [cited 2022Jan.20];56(1):28-. Available from: http://www.matstud.org.ua/ojs/index.php/matstud/article/view/245
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Articles