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

TL;DR

This paper investigates the boundedness of the Hardy-Littlewood maximal operator in variable Lebesgue spaces with exponents in the BLO^{1/ ext{log}} class, providing new counterexamples and clarifying the relationship with BMO^{1/ ext{log}}.

Contribution

It constructs specific variable exponents in BLO^{1/ ext{log}} for which the maximal operator is not bounded, extending previous results related to BMO^{1/ ext{log}}.

Findings

Existence of exponents in BLO^{1/ ext{log}} with unbounded maximal operator

Counterexamples showing boundedness does not always hold in BLO^{1/ ext{log}}

Clarification of the relationship between BMO^{1/ ext{log}} and BLO^{1/ ext{log}} classes

Abstract

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