Weighted bounds for variational Fourier series

Yen Do, Michael Lacey
2012 Studia Mathematica  
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
more » ... f a variational inequality of Lepingle.
doi:10.4064/sm211-2-4 fatcat:duvjfjdggrb6vf35y5dlz4zoje