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It is well known that if Hardy-Littlewood maximal operator is bounded inspace $L^{p(\cdot)}[0;1]$ then $1/p(\cdot)\in BMO^{1/\log}$. On the other handif $p(\cdot)\in BMO^{1/\log},$ ($1<p_{-}\leq p_{+}<\infty$), then there exists$c>0$ such that Hardy-Littlewood maximal operator is bounded in$L^{p(\cdot)+c}[0;1].$ Also There exists exponent $p(\cdot)\in BMO^{1/\log},$($1<p_{-}\leq p_{+}<\infty$) such that Hardy-Littlewood maximal operator is notbounded in $L^{p(\cdot)}[0;1]$. In the present paper we construct exponent$p(\cdot),$ $(1<p_{-}\leq p_{+}<\infty)$, $1/p(\cdot)\in BLO^{1/\log}$ suchthat Hardy-Littlewood maximal operator is not bounded in $L^{p(\cdot)}[0;1]$.

Hardy-Littlewood Maximal Operator And $BLO^{1/\log}$ Class of Exponents