The Variation of the Fractional Maximal Function of a Radial Function

Abstract

In this paper we study the regularity of the noncentered fractional maximal operator M β . As the main result, we prove that there exists C(n, β) such that if q = n/(n − β) andThe corresponding result was previously known only if n = 1 or β = 0. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all f ∈ W 1,1 (R n ).

Regularity of maximal functions on Hardy–Sobolev spaces

“…In the same spirit of , Corollary implies that under stronger conditions than just , which sheds new light on the question if one might have for general .…”

“…The first work in this direction is due to Tanaka [27], who studied the case of ϕ( Recently, Luiro [21] proved that inequality (1.8) is true in any dimension for the uncentered Hardy-Littlewood maximal function, provided one considers only radial functions. Later Luiro and Madrid [22] extended the radial paradigm to the uncentered fractional Hardy-Littlewood maximal function. As a straightforward consequence of Theorem 1, we obtain partial progress toward the understanding of the W 1,1 scenario.…”

Sobolev regularity of polar fractional maximal functions

“…The progress in this problem (for general β) is restricted to the uncentered case. In that case it has been settled affirmatively only in dimension d = 1, according to the already mentioned work [8], and in the radial case [21]. For β ≥ 1, Question 3 has been settled affirmatively in every dimension due to an smoothing property in Okboth centered and uncentered cases (see [8]).…”

“…Concerning to the polar case, the difficulties that Carneiro and the author faced in [6] also appear (in different ways) when dealing with this question. Our first theorem is to get the analogue of the main result of [21] in this context. We go further in the methods already developed in [6] in order to adapt the proof of [21].…”

Regularity of Maximal Operators: Recent Progress and Some Open Problems

“…In the higher dimensional case, partial progress on Question 7 was obtained by Luiro and Madrid in the recent work [29]. They considered the uncentered fractional maximal operator M β acting on radial functions.…”