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
We would like to point out that Xiaochun Li [12] has some unpublished results about weighted estimates for the bilinear Hilbert transform.
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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
Keywords
- Fourier–Walsh series
- Pointwise convergence
- Weighted estimates
- Rubio de Francia inequality
- Variation norms
- Lepingle inequality