For 1 < p < ∞ and for weight w in , we show that the r-variation of the Fourier sums of any function f in 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-2-4,
author = {Yen Do and Michael Lacey},
title = {Weighted bounds for variational Fourier series},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {153-190},
zbl = {1266.42031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-2-4}
}
Yen Do; Michael Lacey. Weighted bounds for variational Fourier series. Studia Mathematica, Tome 209 (2012) pp. 153-190. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm211-2-4/