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

In the present paper we investigate some geometrical properties of the norms in Banach function spaces. Particularly there is shown that if exponent $1\mathrm{/}p(\cdot )$ belongs to $BL{O}^{1\mathrm{/}\log }$ then for the norm of corresponding variable exponent Lebesgue space we have the following lower estimate $${\parallel \sum {\chi }_Q\mathrm{\parallel }f{\chi }_Q{\mathrm{\parallel }}_{p(\cdot )}\mathrm{/}\mathrm{\parallel }{\chi }_Q{\mathrm{\parallel }}_{p(\cdot )}\parallel }_{p(\cdot )}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{p(\cdot )}$$ 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 $$\mathrm{\parallel }f{\mathrm{\parallel }}_{p(\cdot )}\le C{\parallel \sum {\chi }_Q\mathrm{\parallel }f{\chi }_Q{\mathrm{\parallel }}_{p(\cdot )}\mathrm{/}\mathrm{\parallel }{\chi }_Q{\mathrm{\parallel }}_{p(\cdot )}\parallel }_{p(\cdot )}\mathrm{.}$$