This is an announcement for the paper "On the upper and lower estimates of norms in variable exponent spaces" by Tengiz Kopaliani, Nino Samashvili and Shalva Zviadadze.
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 $$|f|_{p(\cdot)}\leq C\left|\sum \chi_{Q}|f\chi_{Q}|_{p(\cdot)}/|\chi_{Q}|_{p(\cdot)}\right|_{p(\cdot)}. $$
Archive classification: math.FA
Mathematics Subject Classification: 42B35, 42B20, 46B45, 42B25
Remarks: 13 pages
Submitted from: sh.zviadadze@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1411.3461
or