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

For a real-valued function $f$ on a metric measure space $(X,d,μ)$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm': $Mf(x):=\sup_{r>0}\frac{1}{μ(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|dμ.$ 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 $μ$ is a left Haar-measure (resp. right Haar-measure) are studied.

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