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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 $[0;1]$. 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