On the upper and lower estimates of norms in variable exponent spaces

TL;DR

This paper explores geometric properties of norms in variable exponent Lebesgue spaces, establishing lower bounds under certain conditions and constructing examples that lack corresponding upper bounds.

Contribution

It provides new lower estimate results for norms in variable exponent spaces and constructs examples showing the absence of certain upper estimates.

Findings

Established lower estimate for norms when $1/p(ullet)$ is in $BLO^{1/ ext{log}}$

Constructed variable exponent spaces without the upper estimate property

Identified geometric conditions affecting norm estimates in variable exponent spaces

Abstract

In the present paper we investigate some geometrical properties of the norms in Banach function spaces. Particularly there is shown that if exponent $1/p(\cdot)$ belongs to $BLO^{1/\log}$ then for the norm of corresponding variable exponent Lebesgue space we have the following lower estimate $$\left\|\sum \chi_{Q}\|f\chi_{Q}\|_{p(\cdot)}/\|\chi_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}\leq C\|f\|_{p(\cdot)}$$ where $\{Q\}$ defines disjoint partition of $[0;1]$. Also we have constructed variable exponent Lebesgue space with above property which does not possess following upper estimation $$\|f\|_{p(\cdot)}\leq C\left\|\sum \chi_{Q}\|f\chi_{Q}\|_{p(\cdot)}/\|\chi_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}. $$