TY - JOUR
AU - Bekbayev, N.T.
AU - Tulenov, K.S.
PY - 2020/06/26
Y2 - 2024/10/12
TI - The non-commutative hardy-littlewood maximal operator on non-commutative lorentz spaces
JF - Journal of Mathematics, Mechanics and Computer Science
JA - JMMCS
VL - 106
IS - 2
SE - Mathematics
DO - 10.26577/JMMCS.2020.v106.i2.03
UR - https://bm.kaznu.kz/index.php/kaznu/article/view/760
SP - 31-38
AB - <p>In this work we study the non-commutative Hardy-Littlewood maximal operator on Lorentz spaces of $\tau$-measurable operators.<br>Non-commutative maximal inequalities were studied, in particular, in \cite{MJ1, JQ, TM}. Another version of the (non-commutative)<br>Hardy-Littlewood maximal operator was introduced by T. Bekjan \cite{TB}. Later J. Shao investigated the Hardy-Littlewood maximal operator<br>on non-commutative Lorentz spaces associated with finite atomless von Neumann algebra (see \cite{Sh}). Namely, for an operator $T$<br>affiliated with a semi-finite von Neumann algebra $\mathcal{M},$ the Hardy-Littlewood maximal operator of $T$ is defined by<br>$$MA(x)=\sup\limits_{r>0}\frac{1}{\tau\left(E_{[x-r, x+r]}\left(|A|\right)\right)}\tau\left(|A|E_{[x-r, x+r]}\left(|A|\right)\right),<br>\,\,x\geq0.$$<br>While the classical Hardy-Littlewood maximal operator of a Lebesgue measurable function $f:\mathbb{R}\rightarrow\mathbb{R}$, denoted by<br>$Mf(x)$, is defined as<br>$$Mf(x)=\sup\limits_{r>0}\frac{1}{m([x-r, x+r])}\int\limits_{[x-r,x+r]}|f(t)|dt,$$<br>where $m$ is a Lebesgue measure on $(-\infty,\infty)$ \cite{SW}. In view of spectral theory, $|A|$ is represented as<br>$$|A|=\int_{\sigma(|A|)}tdE_{t},$$<br>and $MA(|A|)$ is represented as $MA(x).$ Thus, for the operator $A,$ Bekjanâ€™s consideration is that $MA(|A|)$ is defined as the operator<br>analogue of the Hardy-Littlewood maximal operator in the classical case. Our purpose is to investigate the non-commutative<br>Hardy-Littlewood maximal operator $M$ in the sense of T. Bekjan (see \cite{TB}).<br>In particular, we obtain boundedness of the non-commutative Hardy-Littlewood maximal operator in non-commutative Lorentz spaces.</p>
ER -