Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 5 2 2024 04 01 A modification of Hardy-Littlewood maximal-function on Lie groups 143 149 5157 10.22060/ajmc.2023.22259.1147 EN Maysam Maysami Sadr Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran Journal Article 2023 03 10 For a real-valued function $f$ on a metric measure space $(X,d,\mu)$ the Hardy-Littlewood centered-ball maximal-function of $f$ is given by the `supremum-norm':<br />$$Mf(x):=\sup_{r>0}\frac{1}{\mu(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|d\mu.$$<br />In this note, we replace the supremum-norm on parameters $r$ by $\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$ converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$ is a Lie group, $d$ is a left-invariant metric, and $\mu$ is a left Haar-measure (resp. right Haar-measure) are studied. https://ajmc.aut.ac.ir/article_5157_baa11da9a85053a25ed30948c0b7ae4c.pdf