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Weighted bounds for variational Fourier series

By Yen Do and Michael Lacey


For 1<p<infty and for weight w in A_p, we show that the r-variation of the Fourier sums of any function in L^p(w) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality of Lepingle.Comment: 31 pages. v2: Minor changes. To appear in Studia Mat

Topics: Mathematics - Classical Analysis and ODEs
Year: 2012
DOI identifier: 10.4064/sm211-2-4
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