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Article overview
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On a discrete version of Tanaka's theorem for maximal functions | | Jonathan Bober
; Emanuel Carneiro
; Kevin Hughes
; Lillian B. Pierce
; | | Date: |
17 May 2010 | | Abstract: | In this paper we prove a discrete version of Tanaka’s Theorem cite{Ta} for
the Hardy-Littlewood maximal operator in dimension $n=1$, both in the
non-centered and centered cases. For the discrete non-centered maximal operator
$wM $ we prove that, given a function $f: o R$ of bounded variation,
$$Var(wM f) leq Var(f),$$ where $Var(f)$ represents the total variation of
$f$. For the discrete centered maximal operator $M$ we prove that, given a
function $f: o R$ such that $f in ell^1()$, $$Var(Mf) leq C
|f|_{ell^1()}.$$ This provides a positive solution to a question of Hajl
asz and Onninen cite{HO} in the discrete one-dimensional case. | | Source: | arXiv, 1005.3030 | | Services: | Forum | Review | PDF | Favorites | |
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