Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-11T02:13:01.019Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal Operators and Cantor Sets

Published online by Cambridge University Press:  20 November 2018

Kathryn E. Hare
Kathryn E. Hare*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, email: kehare@uwaterloo.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on ${{L}^{2}}$ if the Cantor set has positive Hausdorff dimension.

Keywords

42B2543A46maximal functionCantor setlacunary set
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 3 , 01 September 2000 , pp. 330 - 342
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Arutyunyants, G., On the boundedness of certain maximal and multiplier operators. Preprint, 1996.Google Scholar
[2] Cabrelli, C., Hare, K. and Molter, U., Sums of Cantor sets. Ergodic Theory Dynamical Systems 17 (1997), 1299–1313.Google Scholar
[3] Cordoba, A., The Kakeyamaximal function and the spherical summation multipliers. Amer. J.Math. 99 (1977), 1–22.Google Scholar
[4] Cordoba, A. and Fefferman, R., On differentiation of integrals. Proc. Nat. Acad. Sci. USA 74 (1977), 2211–2213.Google Scholar
[5] deGuzman, M., Real variable methods in Fourier Analysis. Mathematics Studies 46, North Holland, 1981.Google Scholar
[6] Fefferman, C., Themultiplier problemfor the ball. Ann. of Math. 94 (1971), 330–336.Google Scholar
[7] Katz, N. H., Counterexamples for maximal operators over Cantor sets of directions.Math. Res. Lett. 3 (1996), 527–536.Google Scholar
[8] Nagel, A., Stein, E. and Wainger, S., Differentiation in lacunary directions. Proc. Nat. Acad. Sci.USA, 75 (1978), 1060–1062.Google Scholar
[9] Sjogren, P. and Sjolin, P., Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets. Ann. Inst. Fourier (Grenoble) 31 (1981), 157–175.Google Scholar
[10] Stromberg, J., Maximal functions for rectangles with given directions. Thesis, Mittag-Leffler Inst., Sweden, 1976.Google Scholar
[11] Vargas, A., A remark on a maximal function over a Cantor set of directions. Rend. Circ. Mat. Palermo 44 (1995), 273–282.Google Scholar