A modification of Hardy-Littlewood maximal function on Lie groups
| Autor: | Maysam Maysami Sadr |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2024 |
| Předmět: | |
| Zdroj: | AUT Journal of Mathematics and Computing, Vol 5, Iss 2, Pp 143-149 (2024) |
| Druh dokumentu: | article |
| ISSN: | 2783-2449 2783-2287 |
| DOI: | 10.22060/ajmc.2023.22259.1147 |
| Popis: | 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. |
| Databáze: | Directory of Open Access Journals |
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