Regularity of Maximal Operators: Recent Progress and Some Open Problems

Abstract

This is an expository paper on the regularity theory of maximal operators, when these act on Sobolev and BV functions, with a special focus on some of the current open problems in the topic. Overall, a list of fifteen research problems is presented.

Variation of the dyadic maximal function

“…Note that V = C d var satisfies ( 5) and ( 6), but the whole statement becomes trivial because then (7) is already what we want to prove. Still, this shows that the existence of a V as in Claim 5.1 is actually equivalent to (4).…”

“…The most promising candidate is V (f ) = max{var Mf, C d var f }. By [27] it satisfies (6) and it clearly also satisfies (7). In order to prove (5), by the subadditivity of var it suffices to prove that there is a C d such that var M(g + h) ≤ var Mg + C d var h…”

“…It remains unanswered, but there is partial progress, similarly as for the Hardy-Littlewood maximal operator; see for example [5,6,10,24]. For more background information on the regularity of maximal operators there is a survey [7] by Carneiro.…”

Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

“…This continuity property was investigated by Luiro [25] and later extensions were given in [26,8]. More interesting works related to this topic may be found in [2,6,7,9,20,22,23], see also the nice recent survey paper given by Carneiro in [5].…”

The Variation of the Uncentered Maximal Operator with respect to Cubes

“…More endpoint results are available for related maximal operators, for example convolution maximal operators [CS13,CGR19], fractional maximal operators [KS03, CM17, CM17, BM19, BRS19, Wei21, HKKT15], and discrete maximal operators [CH12], as well as maximal operators on different spaces, such as in the metric setting [KT07] and on Hardy-Sobolev spaces [PPSS18]. For more background information on the regularity of maximal operators there is a survey [Car19] by Carneiro. Local regularity properties of the maximal function, which are weaker than the gradient bound of the maximal operator have also been studied [HM10,ACPL12]. The question whether the maximal operator is a continuous operator on the gradient level is even more difficult to answer than its boundedness because the maximal operator is not linear.…”