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

Abstract

Abstract. 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 DM f L 1 (I) ≤ |Df |(I). This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.

Endpoint regularity of discrete multisublinear fractional maximal operators associated with ℓ 1 $\ell^{1}$ -balls

“…Tanaka [ 14 ] first proved that is weakly differentiable and satisfies if . The above result was later refined by Aldaz and Pérez Lázaro [ 15 ] who showed that if f is of bounded variation on , then is absolutely continuous and where denotes the total variation of f . The above result directly yields ( 1.1 ) with constant (see also [ 16 ]).…”

Regularity and continuity of commutators of the Hardy–Littlewood maximal function

“…This question was investigated by Luiro and later extensions were given in . More interesting works related to this topic may be found in .…”

Regularity for commutators of the local multilinear fractional maximal operators

“…Since we do not have sublinearity for the weak derivatives of the Hardy-Littlewood maximal function, the continuity of M : W ,p (R n ) → W ,p (R n ) for < p < ∞ is a rmatively a nontrivial issue, which was addressed by Luiro [23] and later extensions were given in [24]. We can consult [2,4,5,7,25,26] for the endpoint Sobolev regularity of maximal operators, as well as [17,20] for the regularity properties of maximal operators on other smooth function spaces, such as Triebel-Lizorkin spaces, fractional Sobolev spaces and Besov spaces. It should be pointed out that the commutativity with translations for maximal operators plays a key role in deducing the boundedness of maximal operators on W ,p (R n ).…”