Hardy--Littlewood maximal operators on certain manifolds with bounded geometry
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917927807090688 |
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| author | Meda, Stefano Pigola, Stefano Setti, Alberto G. Veronelli, Giona |
| author_facet | Meda, Stefano Pigola, Stefano Setti, Alberto G. Veronelli, Giona |
| contents | In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under mild assumptions, $L^p$ estimates for the centred operator are ``stable'' under conformal changes of the metric, and prove sharp~$L^p$ estimates for the centred operator on Riemannian models with pinched negative scalar curvature. Furthermore, we prove that the centred operator is of weak type $(1,1)$ on the connected sum of two space forms with negative curvature, whereas the uncentred operator is, perhaps surprisingly, bounded only on $L^\infty$. We also prove that if two locally doubling geodesic metric measure spaces enjoying the uniform ball size condition are strictly quasi-isometric, then they share the same boundedness properties for both the centred and the uncentred maximal operator. Finally, we discuss some $L^p$ mapping properties for the centred operator on a specific Riemannian surface introduced by Strömberg, providing new interesting results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hardy--Littlewood maximal operators on certain manifolds with bounded geometry Meda, Stefano Pigola, Stefano Setti, Alberto G. Veronelli, Giona Functional Analysis Differential Geometry 42B25, 53C20 In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under mild assumptions, $L^p$ estimates for the centred operator are ``stable'' under conformal changes of the metric, and prove sharp~$L^p$ estimates for the centred operator on Riemannian models with pinched negative scalar curvature. Furthermore, we prove that the centred operator is of weak type $(1,1)$ on the connected sum of two space forms with negative curvature, whereas the uncentred operator is, perhaps surprisingly, bounded only on $L^\infty$. We also prove that if two locally doubling geodesic metric measure spaces enjoying the uniform ball size condition are strictly quasi-isometric, then they share the same boundedness properties for both the centred and the uncentred maximal operator. Finally, we discuss some $L^p$ mapping properties for the centred operator on a specific Riemannian surface introduced by Strömberg, providing new interesting results. |
| title | Hardy--Littlewood maximal operators on certain manifolds with bounded geometry |
| topic | Functional Analysis Differential Geometry 42B25, 53C20 |
| url | https://arxiv.org/abs/2502.13109 |