On the extension of the Reverse H\"older Inequality for power functions on the real axis

TL;DR

This paper investigates the relationship between the Reverse H"older Inequality norms of functions on the positive real axis and their even extensions to the entire real line, providing bounds and asymptotic sharpness results.

Contribution

It introduces bounds for the ratio of norms between functions on  and their even extensions, with exact results for power functions and asymptotic sharpness for general functions.

Findings

Derived an upper estimate for the ratio of norms.

Established the exact increase of the norm for power functions.

Proved asymptotic sharpness of the bounds.

Abstract

We consider the class of all non-negative on $\mathbb{R_+}$ functions such that each of them satisfies the Reverse H\"older Inequality uniformly over all intervals with some constant the minimum value of which can be regarded as the corresponding "norm" of a function. We compare this "norm" with the "norm" of an even extension of a function from $\mathbb{R_+}$ on $\mathbb{R}.$ In this paper the upper estimate for the ratio of such "norms" has been obtained. In the particular case of power functions on $\mathbb{R_+}$ the precise value of the increase of the "norm" of its even extension is given. This value is the lower estimate for the analogous one in the case of arbitrary functions. It has been shown that the obtained upper and lower estimates for the general case are asymptotically sharp.