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

Abstract

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

6 pages