Weighted bounds for variational Walsh-Fourier series

2011/12/08 by Michael T. Lacey, Lacey, Michael T., Yen Do +1
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #math.CA

paper · pdf · doi:10.48550/arxiv.1112.1744

v2: A major revision of v1, which only considered the pointwise convergence, not the finer variational information

openalex publication_date 2011/12/08 · arxiv created 2012/02/12 · arxiv updated 2012/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) 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 Lepingle.

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