Derivative bounds for fractional maximal functions

Abstract

In this paper we study the regularity properties of fractional maximal operators acting on BVfunctions. We establish new bounds for the derivative of the fractional maximal function, both in the continuous and in the discrete settings.Question A. (Haj lasz and Onninen [10]) Is the operator f → |∇M f | bounded from W 1,1 (R d ) to L 1 (R d )?A standard dilation argument reveals the true nature of this question: whether the variation of the maximal function is controlled by the variation of the original function, i.e. if we haveProgress on this problem has been restricted to dimension d = 1. For the uncentered maximal operator (defined similarly as in (1.1), with the supremum taken over all balls containing the point x in its closure), which we denote here by M , Tanaka [22] showed that if f ∈ W 1,1 (R) then M f is weakly differentiable andIn the case 1 ≤ β < d, Question B admits a positive answer, which follows from the main result of Kinnunen and Saksman in their aforementioned work [13]. In fact, [13, Theorem 3.1] states the following regularizing effect: if f ∈ L r (R d ) with 1 < r < d and 1 ≤ β < d/r, then M β f is weakly differentiable and C(d, β) is a universal constant. In our case, given 1 ≤ β < d and f ∈ W 1,1 (R d ), by the Sobolev embedding we have f ∈ L p * (R d ), where p * = d/ (d − 1), and hence f ∈ L r (R d ) for any 1 ≤ r ≤ p * . We may choose r with 1 < r < d such that 1 ≤ β < d/r and hence (1.5) holds.ThenQuestion B (in the case 0 ≤ β < 1) is the main motivation for this work. The presence of the fractional part introduces additional difficulties as we shall see in the course of the paper (e.g. one does not necessarily have M β (f )(x) ≥ |f (x)| a.e.). Here we give a positive answer to Question B in dimension d = 1 for the uncentered

Regularity of Maximal Operators: Recent Progress and Some Open Problems

“…The problem of finding the sharp constant in this inequality is certainly an interesting one. The strategy of [9] to prove Theorem 5.2 in the pure fractional case β > 0 is very different from that of the proof of Theorem 3.1. While in the proof of Theorem 3.1 the essential idea is to prove that the maximal function does not have any local maxima in the set where it disconnects from the original function, in the fractional case β > 0, the mere notion of the disconnecting set is ill-posed, since one does not necessarily have M β (f )(x) ≥ |f (x)| a.e.…”

“…This shows that the map f → |∇M β f | is bounded from W 1,1 (R d ) to L q (R d ) in this case. We are thus left with the following endpoint question, first posed in [9]. |f…”

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

“…Inequality (14) is not optimal, and it was asked in [29] whether the sharp constant for inequality (14) is in fact = 2; this question was resolved in the affirmative by Madrid in [31]. Later on, the above results were extended to a fractional case in [17,32,33], to a one-sided case in [7], and to a high dimensional case in [34]. For other interesting works we can consult [19,20,[35][36][37].…”

“…Notice that the constant = 1 in inequality (2) is sharp. Recently, inequality (2) was extended to a fractional setting in [17,Theorem 1] and to a multisublinear fractional setting in [18, Theorems 1.3-1.4]. In the centered setting, Kurka [12] showed that if is of bounded variation on R, then inequality (2) holds for M (with constant = 240, 004).…”

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

“…This question was investigated by Luiro [28] and later extensions were given in [7,21,29]. More interesting works related to this topic may be found in [2,5,6,8,13,22,[25][26][27].…”