A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

On approximate differentiability of the maximal function

“…Since the maximal function is not bounded in L 1 , there is no apparent reason to expect any kind of boundedness of the maximal function in W 1,1 (R n ). However, Tanaka [25] proved that in the one dimensional case the noncentered maximal function of f ∈ W 1,1 (R) belongs locally to W 1,1 (R). Since that time it has been an open problem to extend Tanaka's result to the case of the Hardy-Littlewood maximal function and to find analogous results in the higher dimensional case; cf.…”

“…Let us also mention that the result of Kinnunen [10] has been applied and generalized by many authors ( [2], [3], [6], [7], [8], [11], [12], [14], [15], [16], [17], [18], [20], [21], [25]). …”

On the variation of the Hardy-Littlewood maximal function

“…The question had been already answered positively in the non-centered one-dimensional case by Tanaka [15]. This result was sharpened later by Aldaz and Pérez Lázaro [1] who proved that, for an arbitrary f : R → R of bounded variation, its non-centered maximal function M f is absolutely continuous and…”

“…Let x 1 < x 2 < · · · < x l be given. We want to show that In this section, we follow methods from [1] and [15]. We recall that a function f : A ⊂ R → R is said to have Lusin's property (N) (or is called an N-function) on A if, for every set N ⊂ A of measure zero, f (N) is also of measure zero.…”

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

“…A crucial question was posed by Hajłasz and Onninen in [2]: Is the operator → |∇M | bounded from 1,1 (R ) to 1 (R )? A complete solution was achieved only in dimension = 1 in [11][12][13][14] and partial progress on the general dimension ≥ 2 was given by Hajłasz and Malý [15] and Luiro [16]. Tanaka [14] first observed that if ∈ 1,1 (R), thenM is weakly differentiable and…”