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

It is well known that if Hardy-Littlewood maximal operator is bounded in space $L^{p(\cdot )}[0;1]$ then $1\mathrm{/}p(\cdot )\in BM{O}^{1\mathrm{/}\log }$. On the other hand if $p(\cdot )\in BM{O}^{1\mathrm{/}\log },$ ($1<p_{-}\le p_{+}<\mathrm{\infty }$), then there exists $c>0$ such that Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot )+c}[0;1]\mathrm{.}$ Also There exists exponent $p(\cdot )\in BM{O}^{1\mathrm{/}\log },$ ($1<p_{-}\le p_{+}<\mathrm{\infty }$) such that Hardy-Littlewood maximal operator is not bounded in $L^{p(\cdot )}[0;1]$. In the present paper we construct exponent $p(\cdot ),$ $(1<p_{-}\le p_{+}<\mathrm{\infty })$, $1\mathrm{/}p(\cdot )\in BL{O}^{1\mathrm{/}\log }$ such that Hardy-Littlewood maximal operator is not bounded in $L^{p(\cdot )}[0;1]$.