Weighted bounds for variational Fourier series

TL;DR

This paper establishes weighted bounds for the r-variation of Fourier sums in L^p(w) spaces, extending classical results to weighted settings and demonstrating the necessity of the variation exponent's dependence on the weight.

Contribution

It provides a weighted extension of the variational Carleson theorem, showing finiteness of Fourier sum variation for functions in weighted L^p spaces and adapting phase plane analysis.

Findings

Finite a.e. r-variation for Fourier sums in weighted L^p spaces

Dependence of variation exponent on the weight w

Extension of variational inequalities to weighted contexts

Abstract

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 inequality of Lepingle.