Corrigendum to “On a discrete version of Tanaka’s theorem for maximal functions”

By Jonathan W. Bober, Emanuel Carneiro, Kevin A. Hughes, L. Pierce. 2010

70 citations recorded by the source corpus.

In this paper we prove a discrete version of Tanaka's Theorem (19) for the Hardy-Littlewood maximal operator in dimension n = 1, both in the non-centered and centered cases. For the non-centered maximal operator f we prove that, given a function f : Z → R of bounded variation, Var(f Mf) ≤ Var(f), where Var(f) represents the total variation of f. For the centered maximal operator M we prove that, given a function f : Z → R such that f ∈ l 1 (Z), Var(Mf) ≤ Ckfkl1(Z).

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/bc2808042746ccca3418153b3b43422039a0403a

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In Applied Mathematics

Citations in the corpus

On the variation of maximal operators of convolution type (2013, background): “Partial progress was achieved in the discrete setting in the work [3] for dimension d = 1 and in the work [4] for general dimension d > 1.”
On the variation of the Hardy-Littlewood maximal function (2012, unknown): “ Theorem 1.2. Let f: R → be a function of bounded variation. Then VarMf≤ CVarf for a universal constant C. Question 1.1 and the validity of Theorem 1.2 were already studied in the discrete setting in [3]. In the present paper, we do not care how small the constant Cmay be. It is a plausible hypothes”
Derivative bounds for fractional maximal functions (2015, methodology): “8The local extrema are defined slightly differently in [4, 6], but used with the meaning stated here.”
Endpoint Sobolev and BV continuity for maximal operators, II (2017, methodology): “The notation and terminology used in this paper are classical in the theory, and in particular align with our previous works [4, 8, 10] on the subject.”
On the endpoint regularity of discrete maximal operators (2013, unknown): “This has only been proved in dimension d = 1 for the non-centered maximal operator (see [2]) with sharp constant C = 1 (i.”