Weighted bounds for variational Walsh-Fourier series
  For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the   Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) This strengthens a result of Hunt-Young and is a weighted extension of a variation norm Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lepingle. 
Michael T. Lacey and Yen Do
math.CA