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## Weighted bounds for variational Fourier series

### Tom 211 / 2012

Studia Mathematica 211 (2012), 153-190 MSC: Primary 42B20; Secondary 42B25, 42B35. DOI: 10.4064/sm211-2-4

#### Streszczenie

For $1< p< \infty$ and for weight $w$ in $A_p$, we show that the $r$-variation of the Fourier sums of any function $f$ 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 Lépingle.

#### Autorzy

• Yen DoDepartment of Mathematics
Yale University
New Haven, CT 06511, U.S.A.
e-mail
• Michael LaceySchool of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332, U.S.A.
e-mail

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