Variation of the uncentered maximal characteristic function

Weigt, Julian

doi:10.4171/rmi/1312

Cited by 9 publications

(5 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Theorem 6.2, we would of course like to prove that M u ∈ BV loc (R d ), instead of assuming this. For sets of finite perimeter, such a better result is possible due to a very recent result of Weigt [12]. We restate Theorem 1.3, which is our third main theorem:…”

Section: Sobolev Propertymentioning

confidence: 81%

“…In a potential breakthrough toward a solution to the W 1,1 -problem, Weigt [12] has shown very recently that for a set of finite perimeter E ⊂ Ω, we have M Ω ½ E ∈ BV loc (Ω) such that |DM Ω ½ E |(Ω) is at most a constant times |D½ E |(Ω). We can utilize this result and go a step further to the desired Sobolev regularity at least in the global case Ω = R d , as follows.…”

Section: Introductionmentioning

confidence: 99%

“…We also show that if the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of Weigt [12], we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On the regularity of the maximal function of a BV function

Lahti¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

Section: Sobolev Propertymentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the regularity of the maximal function of a BV function

Lahti¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Toward answering this question, partial results have been proved by many authors in e.g. [2,34,38,55]; often one considers the non-centered maximal function, where the supremum is taken over balls containing x. In particular, Haj lasz-Malý [23] showed the Hardy-Littlewood maximal function to be approximately differentiable a.e.…”

Section: Recall the Definition Of The Hardy-littlewood Maximal Functi...mentioning

confidence: 99%

Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

Lahti¹

2022

Preprint

View full text Add to dashboard Cite

We introduce a generalized version of the local Lipschitz number lip u, and show that it can be used to characterize Sobolev functions u ∈ W 1,p loc (R n ), 1 ≤ p ≤ ∞. This concept turns out to be fruitful for studying, and for establishing new connections between, a wide range of topics including fine differentiability, Rademacher's theorem, regularity of maximal functions, quasiconformal mappings, Alberti's rank one theorem, as well as generalizations to metric measure spaces.

show abstract

“…the subspace of W 1,1 (R d ) consisting of radial functions. There are also a couple of promising new results by J. Weigt, solving the total variation version of this question for characteristic functions of sets of finite perimeter [30], and its analogue for the dyadic maximal operator [31]. Related works in this topic include [7,8,11,13,16,19,25,26,27].…”

mentioning

confidence: 99%

Sunrise strategy for the continuity of maximal operators

Carneiro¹,

González-Riquelme²,

Madrid³

2020

Preprint

View full text Add to dashboard Cite

In this paper we address the W 1,1 -continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into "lateral" maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy-Littlewood maximal operator M acting on W 1,1 rad (R d ), the subspace of W 1,1 (R d ) consisting of radial functions. In dimension d ≥ 2 it was recently established by H. Luiro that the map f → ∇ M f), and we show that such map is also continuous. Further applications of the sunrise strategy in connection with the W 1,1 -continuity problem include nontangential maximal operators on R d acting on radial functions when d ≥ 2 and general functions when d = 1, and the uncentered Hardy-Littlewood maximal operator on the sphere S d acting on polar functions when d ≥ 2 and general functions when d = 1.

show abstract

Variation of the uncentered maximal characteristic function

Abstract: Let M be the uncentered Hardy-Littlewood maximal operator, or the dyadic maximal operator, and let d 1. We prove that for a set E R d of finite perimeter, the bound var M1 E Ä C d var 1 E holds. We also prove this for the local maximal operator.

Cited by 9 publications

References 14 publications

On the regularity of the maximal function of a BV function

On the regularity of the maximal function of a BV function

Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

Sunrise strategy for the continuity of maximal operators

Contact Info

Product

Resources

About