A modification of Hardy-Littlewood maximal-function on Lie groups

For a real-valued function $f$ on a metric measure space $(X,d,\mu )$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm’: $$Mf(x):=\underset{r>0}{\sup }\frac{1}{\mu ({\mathcal{B}}_{x,r})}{\int }_{{\mathcal{B}}_{x,r}}\mathrm{\mid }f\mathrm{\mid }d\mu \mathrm{.}$$ 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 $p\to \mathrm{\infty }$. 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.