Study of Bounded Variation on the H-L Nontangential Operator
Frederico Toulson
TL;DR
The paper establishes sharp bounded variation bounds for the one-dimensional Hardy-Littlewood nontangential maximal operator in both discrete and continuous settings, proving that $\mathrm{Var}\,(M^{\alpha}f) \le \mathrm{Var}\,(f)$ for all $\alpha\ge \tfrac{1}{3}$ and $f$ of bounded variation. It generalizes a discrete uncentered result by BCHP to the nontangential (centered) context and provides an elementary alternative proof for the continuous case originally given by Ramos. The core technique relies on peak-based decompositions: segmenting the domain into monotone intervals and bounding the maximal function’s variation by the original function’s variation via well-controlled peaks. This contributes a unified, accessible approach to BV bounds for both discrete and continuous Hardy-Littlewood-type operators and clarifies the critical endpoint $\alpha=\tfrac{1}{3}$ in the nontangential setting.
Abstract
In this paper we present a proof of sharp boundedness of the discrete 1-dimensional Hardy-Littlewood nontangential maximal operator, when the parameter is in the range $[\frac{1}{3},+\infty)$. This generalizes a theorem by Bober, Carneiro, Hughes and Pierce, where they prove the same result for the uncentered version of the maximal operator. We also use analogous ideas to give an alternative proof for the continuous version of the theorem, by Ramos.