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On the centered Hardy-Littlewood maximal operator


Author: Antonios D. Melas
Journal: Trans. Amer. Math. Soc. 354 (2002), 3263-3273
MSC (2000): Primary 42B25
DOI: https://doi.org/10.1090/S0002-9947-02-02900-8
Published electronically: February 20, 2002
MathSciNet review: 1897399
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Abstract: We will study the centered Hardy-Littlewood maximal operator acting on positive linear combinations of Dirac deltas. We will use this to obtain improvements in both the lower and upper bounds or the best constant $C$ in the $L^{1}\rightarrow$ weak $L^{1}$ inequality for this operator. In fact we will show that $\frac{11+\sqrt{61}}{12}=1.5675208...\leq C\leq\frac{5} {3}=1.66...$.


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  • 1. J.M.Aldaz. Remarks on the Hardy-Littlewood maximal function, Proc. of the Royal Society of Edinburgh 128A (1998), no.1, 1-9. MR 99b:42020
  • 2. D.A.Brannan, W.K. Hayman. Research problems in complex analysis, Bull. London Math. Soc. 21 (1989), 1-35. MR 89m:30001
  • 3. A.Bernal. A note on the one-dimensional maximal function, Proc. of the Royal Society of Edinburgh 111A (1989), 325-328. MR 90i:28010
  • 4. H.Carlsson. A new proof of the Hardy Littlewood maximal theorem, Bull. London Math. Soc. 16 (1984), 595-596. MR 86g:42024
  • 5. A.M.Garsia. Topics in almost everywhere convergence, Markkham Publishing Company, Chicago, 1970. MR 41:5869
  • 6. L.Grafakos, S.Montgomery-Smith. Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (1997), no.1, 60-64. MR 98b:42031
  • 7. M. de Guzmán. Real variable methods in Fourier analysis, North-Holland Mathematical Studies 46. Notas de Mathemática (75), 1981. MR 83j:42019
  • 8. J. Manfredi, F. Soria. On a dynamical system relared to estimating the best constant in an inequality of Hardy and Littlewood, unpublished manuscript.
  • 9. M.Trinidad Menarguez, F.Soria. Weak type (1,1) inequalities of maximal convolution operators, Rendiconti del Circolo Mathematico di Palermo (2) 41 (1992), 342-352. MR 94i:42025
  • 10. D.Termini, C.Vitanza. Weighted estimates for the Hardy-Littlewood maximal operator and Dirac deltas, Bull. London Math. Soc. 22 (1990), 367-374. MR 91f:42018

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Additional Information

Antonios D. Melas
Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
Email: amelas@math.uoa.gr

DOI: https://doi.org/10.1090/S0002-9947-02-02900-8
Received by editor(s): March 14, 2000
Received by editor(s) in revised form: June 15, 2001
Published electronically: February 20, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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