Weighted bounds for variational Fourier series
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.