Continuity of the maximal operator in Sobolev spaces

Abstract

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces W 1,p (R n) , 1 < p < ∞. As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

On the continuity of discrete maximal operators in Sobolev spaces

“…We will show that this additional assumption yields the continuity of M * in W 1,p (R n ) , when 1 < p < ∞ . The proof follows essentially the same lines as in [9].…”

“…This definition was introduced in the case of the usual Hardy-Littlewood maximal function in [9]. Here we also make a convention T ∞ f = |f | and 2 −∞ = 0 , thus if M * f (x) = |f (x)|, then 0 ∈ Rf (x).…”

“…[10]. By the main result of [9], the classical Hardy-Littlewood maximal operator M is continuous in W 1,p (R n ), when p > 1. In the case of metric measure spaces, as an example in [2] shows, there is an unexpected obstruction concerning the regularity of M. Indeed, it might even happen that Mf of a Lipschitz continuous function f may fail to be continuous.…”

On the regularity of maximal operators

“…To overcome this difficulty, we adopt here the approach introduced in [3]. For the continuity part we follow the insightful and elegant proof of Luiro in [8]. Again the case n = 1, r = 1 is a new feature.…”

“…In this case the sublinearity of the operator implies continuity. As already mentioned in the Introduction of this paper, the operator M defined in (4.1) is bounded and continuous from [4] and [8]) and the local maximal operator M Ω is bounded from W 1,p (Ω) to W 1,p (Ω) (see [5]). We may ask ourselves if these classical maximal operators preserve other types of convergence, for instance pointwise convergence almost everywhere or weak convergence.…”