Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial... 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 $${p\in (1, 1.03]}$$ , 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2010 by Birkhäuser / Springer Basel AG
Subject
Mathematics; Operator Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics; Potential Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0051-2
Publisher site
See Article on Publisher Site

Abstract

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 $${p\in (1, 1.03]}$$ , 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.

Journal

PositivitySpringer Journals

Published: Mar 24, 2010

References

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