Variation inequalities for the Fejér and Poisson kernels

Abstract

In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $1<p<\infty$, when $\varrho >2$. Counterexamples are given if $\varrho =2$.