A note on the endpoint regularity of the discrete maximal operator Article Swipe

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Bounded function Maximal operator Operator (biology) Combinatorics Mathematics Bounded variation Discrete mathematics Mathematical analysis Gene Chemistry Repressor Transcription factor Biochemistry

Feng Liu , Huoxiong Wu ·

YOU? · · 2017 · Open Access · · DOI: https://doi.org/10.1090/proc/13962 · OA: W2762516999

In this note we study the regularity properties of the discrete maximal operators at endpoint. Precisely, we show that the general discrete centered and non-centered maximal operators are bounded and continuous from $\ell ^1(\mathbb {Z})$ to $\textrm {BV}(\mathbb {Z})$, as well as the non-centered discrete maximal operator maps $\textrm {BV}(\mathbb {Z})\rightarrow \textrm {BV}(\mathbb {Z})$ boundedly under a more restrictive condition, where $\textrm {BV}(\mathbb {Z})$ denotes the set of functions of bounded variation defined on $\mathbb {Z}$. As an immediate consequence, we obtain somewhat unexpected endpoint regularities of the discrete fractional maximal functions.