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': $$Mf(x):=\sup_{r>0}\frac{1}{\mu(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|d\mu.$$ 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.
Maysami Sadr, M. (2024). A modification of Hardy-Littlewood maximal-function on Lie groups. AUT Journal of Mathematics and Computing, 5(2), 143-149. doi: 10.22060/ajmc.2023.22259.1147
MLA
Maysam Maysami Sadr. "A modification of Hardy-Littlewood maximal-function on Lie groups". AUT Journal of Mathematics and Computing, 5, 2, 2024, 143-149. doi: 10.22060/ajmc.2023.22259.1147
HARVARD
Maysami Sadr, M. (2024). 'A modification of Hardy-Littlewood maximal-function on Lie groups', AUT Journal of Mathematics and Computing, 5(2), pp. 143-149. doi: 10.22060/ajmc.2023.22259.1147
VANCOUVER
Maysami Sadr, M. A modification of Hardy-Littlewood maximal-function on Lie groups. AUT Journal of Mathematics and Computing, 2024; 5(2): 143-149. doi: 10.22060/ajmc.2023.22259.1147
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