The variation of the maximal function of a radial function

Abstract

Abstract. It is shown for the non-centered Hardy-Littlewood maximal operator M that DM f 1 ≤Cn Df 1 for all radial functions in 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 .…”

“…There has been a lot of effort in understanding this question in the last few years, as well as the problem of determining the optimal constant in (1.8). 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.…”

Regularity of Maximal Operators: Recent Progress and Some Open Problems

“…This is a weaker notion than that of classical differentiability or weak differentiability. In fact, if a function f is differentiable at a point x 0 then it is approximately differentiable at x 0 and L = ∇f (x 0 ), and similarly, if f is weakly differentiable then it is approximately differentiable and its approximate differential is equal to the weak derivative a.e., see for instance [13, The recent interesting work of Luiro [28] answers Question 1 affirmatively in the case of the uncentered maximal function M and restricted to radial functions f . Theorem 3.5 (Luiro, 2017 -cf.…”

Regularity of fractional maximal functions through Fourier multipliers

“…by Tanaka [38] when n = 1 and Luiro [27] when n ≥ 2 and f is radial. Here M is the non-centred Hardy-Littlewood maximal function.…”