The Hardy--Littlewood maximal operator and BLO1/log class of exponents

It is well known that if the Hardy–Littlewood maximal operator is bounded in the variable exponent Lebesgue space ${L^{p(\,\cdot\,)}[0;1]}$, then ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$. On the other hand, there exists an exponent ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$, ${1<p_{-}\leq p_{+}<\infty}$, such that the Hardy–Littlewood maximal operator is not bounded in ${L^{p(\,\cdot\,)}[0;1]}$. But for any exponent ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$, ${1<p_{-}\leq p_{+}<\infty}$, there exists a constant ${c>0}$ such that the Hardy–Littlewood maximal operator is bounded in ${L^{p(\,\cdot\,)+c}[0;1]}$. In this paper, we construct an exponent ${p(\,\cdot\,)}$, ${1<p_{-}\leq p_{+}<\infty}$, ${1/p(\,\cdot\,)\in\mathrm{BLO}^{1/\log}}$ such that the Hardy–Littlewood maximal operator is not bounded in ${L^{p(\,\cdot\,)}[0;1]}$.