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The Buckley Dyadic Square Function

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Recent Advances in Harmonic Analysis and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 25))

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Abstract

We explore the relationships and differences between the “standard” dyadic square function S(f) and a pointwise smaller square function S b (f) due to Stephen Buckley. In dimension one, S(f)=S b (f), but in higher dimensions, S b (f) can vanish on a cube in which S(f) is everywhere positive, and uniform boundedness of S b (f) does not imply the same of S(f). However, uniform boundedness of S b (f) implies local exponential-square integrability of both S(f) and f. The second implication is surprising because the result of [2] requires S(f)∈L ∞ to infer that f is in the local exponential L 2 class.

AMS Subject Classification (2000): 42B25.

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Fedor Nazarov

  2. Department of Mathematics, University of Vermont, Burlington, VT, 05405, USA

    Michael Wilson

Authors
  1. Fedor Nazarov
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  2. Michael Wilson
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Corresponding author

Correspondence to Michael Wilson .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of South Carolina, Greene Street 1523, Columbia, 29208, South Carolina, USA

    Dmitriy Bilyk

  2. University Park, Department of Mathematics, Florida International University, SW 8th St., Rm 416 11200, Miami, 33199, Florida, USA

    Laura De Carli

  3. , Department of Mathematics, Universtity of Georgia, University of Georgia, Athens, 30602, Georgia, USA

    Alexander Petukhov

  4. Georgia Southern University, PO Box 8093, Statesboro, 30460, Georgia, USA

    Alexander M. Stokolos

  5. , Department of Mathematics, Georgia Institute of Technology, Cherry Street 686, Atlanta, 30332, Georgia, USA

    Brett D. Wick

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Nazarov, F., Wilson, M. (2012). The Buckley Dyadic Square Function. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_23

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