Finely continuously differentiable functions

Lávička, Roman

doi:10.1016/j.exmath.2008.02.002

Cited by 3 publications

(5 citation statements)

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“…In this section we study the fine differentiability of Sobolev and BV functions, as well as of the Hardy-Littlewood maximal function. For some previous studies of fine differentiability and finely harmonic functions, see Gardiner [19,20] and Lávička [42,43,44]. These references study the case p = 2, whereas we will keep working with the 1-fine topology.…”

Section: Fine Differentiability and The Hardy-littlewood Maximal Func...mentioning

confidence: 99%

Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

Lahti¹

2022

Preprint

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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.

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Section: Fine Differentiability and The Hardy-littlewood Maximal Func...mentioning

confidence: 99%

Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

Lahti¹

2022

Preprint

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“…In this case, we shall explain just main ideas and refer to [15] for details. In the connection with the case of m = 2, let us remark that since 1970's B. Fuglede and others have developed very deep theory of finely holomorphic functions, see Vol.…”

Section: Theorem 12 Let M ≥ 2 and U ⊂ R M Be Finely Open Thenmentioning

confidence: 99%

“…In [15], Proposition 2.5 is proved using Theorem 2.2 and B. Fuglede's theorem on absolute continuity of functions of W 1,2 (R m ) along almost every curve.…”

Section: Roman Lávička Aacamentioning

confidence: 99%

A Remark on Fine Differentiability

Lávička

2007

AACA

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“…This paper concerns the notion of fine differentiability of functions of the form f : U → R, where U is a finely open subset of R n . Previous papers dealing with this topic include Mastrangelo [12], Mastrangelo and Dehen [13], Raynaud-Pimenta [14], and Lávička [9,10].…”

Section: Introductionmentioning

confidence: 99%

“…When n = 2, Lávička [9] showed that this is the case by adapting work of Fuglede on finely holomorphic functions. The proof is based on a special property of thin sets in the plane, and in higher dimensions much less is known [10]. In fact, Lávička [9] has pointed out that, in this case, it is not even known whether a function that has zero fine differential at every point of a fine domain must be constant.…”

Section: Introductionmentioning

confidence: 99%