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Transactions of the American Mathematical Society

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On the Hardy-Littlewood maximal function and some applications


Author: C. J. Neugebauer
Journal: Trans. Amer. Math. Soc. 259 (1980), 99-105
MSC: Primary 42B25; Secondary 28A15
MathSciNet review: 561825
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Abstract: With a monotone family $ F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf{R}}^n}$, we associate the Hardy-Littlewood maximal function $ {M_F}f(x)\, = \,{\sup _\alpha }(1/\left\vert {{S_\alpha }} \right\vert)\int_{{S_\alpha }\, + \,x} {\left\vert f \right\vert} $. In general, $ {M_F}$ is not weak type (1.1). However, if we replace in the denominator $ {S_\alpha }$ by $ S_F^ {\ast} \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\} $, and denote the resulting maximal function by $ M_F^ {\ast} $, then $ M_F^ {\ast} $ is weak type (1, 1) with weak type constant 1.


References [Enhancements On Off] (What's this?)

  • [1] Luis A. Caffarelli and Calixto P. Calderón, Weak type estimates for the Hardy-Littlewood maximal functions, Studia Math. 49 (1973/74), 217–223. MR 0335729
  • [2] Calixto P. Calderón, Differentiation through starlike sets in 𝑅^{𝑚}, Studia Math. 48 (1973), 1–13. MR 0330395
  • [3] R. E. Edwards and Edwin Hewitt, Pointwise limits for sequences of convolution operators, Acta Math. 113 (1965), 181–218. MR 0177259
  • [4] M. Guzmán, Differentiation of integrals in $ {{\textbf{R}}^n}$, Lecture Notes in Math., vol. 481, Springer-Verlag, Berlin and New York, 1975.
  • [5] N. M. Rivière, Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243–278. MR 0440268
  • [6] S. Saks, Remark on the differentiability of the Lebesgue indefinite integral, Fund. Math. 22 (1934), 257-261.
  • [7] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0561825-7
Keywords: Hardy-Littlewood maximal function, monotone family, weak type
Article copyright: © Copyright 1980 American Mathematical Society