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Sharp inequalities for maximal functions associated with general measures

Published online by Cambridge University Press:  14 November 2011

L. Grafakos  and
J. Kinnunen
L. Grafakos
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65203, U.S.A. E-mail: loukas@math.missouri.edu
J. Kinnunen
Affiliation:
Department of Mathematics, P.O. Box 4, FIN-00014University of Helsinki, Finland E-mail: Juha.Kinnunen@Helsinki.Fi
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Abstract

Sharp weak type (1,1) and Lp estimates in dimension one are obtained for uncentred maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions, uncentred maximal functions fail to satisfy such estimates. Analogous results for centred maximal functions are given in all dimensions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 717 - 723
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Anderson, K. F.. Weighted inequalities for maximal functions associated with general measures. Trans. Amer. Math. Soc. 326 (1991), 907–20.CrossRefGoogle Scholar
2Bernal, A.. A note on the one-dimensional maximal function. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 325–8.CrossRefGoogle Scholar
3deGuzman, M.. Differentiation of integrals in ℝ″, Lecture Notes in Mathematics 481 (Berlin: Springer, 1975).Google Scholar
4Fefferman, R.. Strong differentiation with respect to measures. Amer. J. Math. 103 (1981), 33–40.CrossRefGoogle Scholar
5Füredi, Z. and Loeb, P. A.. On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc. 121 (1994), 1063–73.CrossRefGoogle Scholar
6Garnett, J.. Bounded analytic functions, Pure and Applied Mathematics 96 (New York: Academic Press, 1981).Google Scholar
7Grafakos, L. and Montgomery-Smith, S.. Best constants for uncentred maximal functions. Bull. London Math. Soc. 29 (1996), 60–4.CrossRefGoogle Scholar
8Sjögren, P.. A remark on the maximal function for measures in ℝ″. Amer. J. Math. 105 (1983), 1231–3.CrossRefGoogle Scholar
9Stein, E. M.. Harmonic Analysis: Real-Variable Theory, Orthogonality, and Oscillatory Integrals (Princeton, NJ: Princeton University Press, 1993).Google Scholar