THE NON-COMMUTATIVE HARDY-LITTLEWOOD MAXIMAL OPERATOR ON NON-COMMUTATIVE LORENTZ SPACES

Abstract

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 operatoron 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 operatoranalogue of the Hardy-Littlewood maximal operator in the classical case. Our purpose is to investigate the non-commutativeHardy-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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Published
2020-06-26
How to Cite
BEKBAYEV, N.T.; TULENOV, K.S.. THE NON-COMMUTATIVE HARDY-LITTLEWOOD MAXIMAL OPERATOR ON NON-COMMUTATIVE LORENTZ SPACES. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 106, n. 2, p. 31-38, june 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/760>. Date accessed: 23 sep. 2020. doi: https://doi.org/10.26577/JMMCS.2020.v106.i2.03.
Keywords Cesaro operator, Hardy-Littlewood maximal operator, Lorentz space