Weighted bounds for variational Fourier series

Yen Do ¹   ; Michael Lacey ²

¹ Department of Mathematics Yale University New Haven, CT 06511, U.S.A.
² School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, U.S.A.

Studia Mathematica, Tome 211 (2012) no. 2, pp. 153-190

Résumé

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.

Détail

DOI : 10.4064/sm211-2-4

Keywords: infty weight r variation fourier sums function finite larger finite constant depending the variation exponent depends necessary strengthens previous work hunt young weighted extension variational carleson theorem oberlin seeger tao thiele wright proof uses weighted adaptation phase plane analysis weighted extension variational inequality pingle