The non-commutative hardy-littlewood maximal operator on non-commutative lorentz spaces
DOI:
https://doi.org/10.26577/JMMCS.2020.v106.i2.03Keywords:
Cesaro operator, Hardy-Littlewood maximal operator, Lorentz spaceAbstract
In this work we study the non-commutative Hardy-Littlewood maximal operator on Lorentz spaces of $\tau$-measurable operators.
Non-commutative maximal inequalities were studied, in particular, in \cite{MJ1, JQ, TM}. Another version of the (non-commutative)
Hardy-Littlewood maximal operator was introduced by T. Bekjan \cite{TB}. Later J. Shao investigated the Hardy-Littlewood maximal operator
on non-commutative Lorentz spaces associated with finite atomless von Neumann algebra (see \cite{Sh}). Namely, for an operator $T$
affiliated with a semi-finite von Neumann algebra $\mathcal{M},$ the Hardy-Littlewood maximal operator of $T$ is defined by
$$MA(x)=\sup\limits_{r>0}\frac{1}{\tau\left(E_{[x-r, x+r]}\left(|A|\right)\right)}\tau\left(|A|E_{[x-r, x+r]}\left(|A|\right)\right),
\,\,x\geq0.$$
While the classical Hardy-Littlewood maximal operator of a Lebesgue measurable function $f:\mathbb{R}\rightarrow\mathbb{R}$, denoted by
$Mf(x)$, is defined as
$$Mf(x)=\sup\limits_{r>0}\frac{1}{m([x-r, x+r])}\int\limits_{[x-r,x+r]}|f(t)|dt,$$
where $m$ is a Lebesgue measure on $(-\infty,\infty)$ \cite{SW}. In view of spectral theory, $|A|$ is represented as
$$|A|=\int_{\sigma(|A|)}tdE_{t},$$
and $MA(|A|)$ is represented as $MA(x).$ Thus, for the operator $A,$ Bekjan’s consideration is that $MA(|A|)$ is defined as the operator
analogue of the Hardy-Littlewood maximal operator in the classical case. Our purpose is to investigate the non-commutative
Hardy-Littlewood maximal operator $M$ in the sense of T. Bekjan (see \cite{TB}).
In particular, we obtain boundedness of the non-commutative Hardy-Littlewood maximal operator in non-commutative Lorentz spaces.
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