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Weighted Bounds for Variational Walsh–Fourier Series

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Abstract

For 1<p<∞ and a weight w∈A p and a function in L p([0,1],w) we show that variational sums with sufficiently large exponents of its Walsh–Fourier series are bounded in L p(w). This strengthens a result of Hunt–Young and is a weighted extension of a variation norm Carleson theorem of Oberlin–Seeger–Tao–Thiele–Wright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lépingle.

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Notes

  1. We would like to point out that Xiaochun Li [12] has some unpublished results about weighted estimates for the bilinear Hilbert transform.

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Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, 06511, USA

    Yen Do

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Michael Lacey

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  1. Yen Do
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  2. Michael Lacey
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Correspondence to Michael Lacey.

Additional information

Communicated by Loukas Grafakos.

Research supported in part by grant NSF-DMS-0635607002.

Research supported in part by grant NSF-DMS 0968499 and a grant from the Simons Foundation (#229596 to Michael Lacey).

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Do, Y., Lacey, M. Weighted Bounds for Variational Walsh–Fourier Series. J Fourier Anal Appl 18, 1318–1339 (2012). https://doi.org/10.1007/s00041-012-9231-8

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  • DOI: https://doi.org/10.1007/s00041-012-9231-8

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