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A characterization of Sobolev spaces via local derivatives

100%
Swanson D.
Colloquium Mathematicum
|
2010
|
tom 119
|
nr 1
157-167
EN
Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function $f ∈ W^{k,p}(Ω)$ possesses an $L^{p}$ derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space $W^{k,p}(Ω)$. Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.
2
Content available remote

Pointwise inequalities and approximation in fractional Sobolev spaces

100%
Swanson D.
Studia Mathematica
|
2002
|
tom 149
|
nr 2
147-174
EN
We prove that a function belonging to a fractional Sobolev space $L^{α,p}(ℝⁿ)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m,λ}(ℝⁿ)$, 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].
3
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Construction of an Uncountable Difference between Φ(B) and $Φ_f(B)$

63%
Campbell J., Swanson D.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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tom 56
|
nr 2
121-130
EN
We construct a set B and homeomorphism f where f and $f^{-1}$ have property N such that the symmetric difference between the sets of density points and of f-density points of B is uncountable.
4
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Fine behavior of functions whose gradients are in an Orlicz space

51%
Malý J., Swanson D., Ziemer W.
Studia Mathematica
|
2009
|
tom 190
|
nr 1
33-71
EN
For functions whose derivatives belong to an Orlicz space, we develop their "fine" properties as a generalization of the treatment found in [MZ] for Sobolev functions. Of particular importance is Theorem 8.8, which is used in the development in [MSZ] of the coarea formula for such functions.
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