Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities

By J. M. Aldaz, J. Lázaro. 2006

126 citations recorded by the source corpus.

We prove that if f: I ⊂ R→ R is of bounded variation, then the uncentered maximal function Mf is absolutely continuous, and its derivative satisfies the sharp inequality ∥DMf∥ L 1(I) < |Df|(I). This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1df7f8709ecd10655ced0207d34549feb819a7e2

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In Applied Mathematics

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Citations in the corpus

Corrigendum to “On a discrete version of Tanaka’s theorem for maximal functions” (2010, methodology): “ of these results, both in the continuous and discrete cases, to hold as well (see the original question by Hajla sz and Onninen [6]). However, the methods of Tanaka [19] and Aldaz and P´erez L´azaro [1] for the one dimensional continuous (uncentered) case, and ours for the discrete (centered and un”
On the variation of maximal operators of convolution type (2013, background): “al estimate (1.2) is known to hold with constant C= 240,004, but the problem with C= 1 remains open. For the onedimensional right (or left) Hardy-Littlewood maximal operator, i.e. when ϕ(x,t) = 1 t χ [0,1](x/t), estimate (1.2) holds with C= 1 due to the work of Tanaka [21]. The sharp bound (1.2) wit”
A stability version of Hölder's inequality (2007, methodology): “nterval containing xand |I| stands for its length. Starting with the paper [Ki], there has been in recent years a growing interest regarding the regularity of the maximal function (cf., for instance, [AlPe] and the references contained therein). Suppose for simplicity that f : R → R is a compactly s”
On the regularity of maximal operators (2008, background): “Other interesting papers related to this topic are [1], [3] and [6].”
On approximate differentiability of the maximal function (2010, background): “…the best of our knowledge there are no known higher dimensional results in the case p = 1, and even in the one dimensional case it is still not known whether the Hardy-Littlewood maximal function (i.e. the centered one) of f ∈ W 1 , 1 ( R ) belongs locally to W 1 , 1 ( R ); see, however, [2], [3].”
Derivative bounds for fractional maximal functions (2015, methodology): “B25, 39A12, 46E35, 46E39. Key words and phrases. Fractional maximal operator, Sobolev spaces, discrete maximal operators, bounded variation. 1 This result was later reﬁned by Aldaz and P´erez La´zaro [2], who showed that if f is of bounded variation then Mff is in fact absolutely continuous and Var ”
Endpoint Sobolev and BV continuity for maximal operators, II (2017, methodology): “r the map f → ∇Mf is bounded from W1,1(Rd) to L1(Rd) was raised by Hajla sz and Onninen in [12] and remains unsolved in its full generality. A complete solution was reached only in dimension d = 1 in [2, 17, 29] and partial progress on the general case d > 1 was obtained by Hajla sz and Maly´ [11”