Weighted bounds for variational Walsh-Fourier series

TL;DR

This paper establishes weighted bounds for the r-variation of Walsh-Fourier sums in L^p(w) spaces, extending previous results and demonstrating the necessity of r depending on the weight w.

Contribution

It provides a weighted extension of the variation norm Carleson theorem for Walsh-Fourier series, using phase plane analysis and a weighted variational inequality.

Findings

Finite r-variation of Walsh-Fourier sums for p in (1,∞) and weights in A_p.

r must depend on the weight w for the bounds to hold.

Extension of previous unweighted and scalar results to weighted settings.

Abstract

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.