Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures
As shown in Aldaz (Bull. Lond. Math. Soc. 39:203-208, 2007), the lowest constants appearing in the weak type (1, 1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this r...
| Authors: | , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2010 |
| Country: | España |
| Institution: | Universidad de La Rioja (UR) |
| Repository: | RIUR. Repositorio Institucional de la Universidad de La Rioja |
| OAI Identifier: | oai:portal.dialnet.es:doc/5bbc6913b750603269e814b8 |
| Online Access: | https://investigacion.unirioja.es/documentos/5bbc6913b750603269e814b8 |
| Access Level: | Open access |
| Keyword: | Maximal operators Radial measures Weak type bounds |
| Summary: | As shown in Aldaz (Bull. Lond. Math. Soc. 39:203-208, 2007), the lowest constants appearing in the weak type (1, 1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p > 1. Furthermore, we improve the previously known bounds for p = 1. Roughly speaking, whenever p ∈(1,1.03], if μ is defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c p,d,μ in the weak type (p, p) inequalities satisfy c p,d,μ ≥ 1.005 d for all d sufficiently large. We also show that exponential increase of the best constants occurs for certain families of doubling measures, and for arbitrarily high values of p. © 2010 Birkhäuser / Springer Basel AG. |
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