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1
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Composition operators on W 1 X are necessarily induced by quasiconformal mappings

100%
Kleprlík L.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1229-1238
EN
Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.
2
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Polynomial asymptotics and approximation of Sobolev functions

88%
Hajłasz P., Kałamajska A.
Studia Mathematica
|
1995
|
tom 113
|
nr 1
55-64
EN
We prove several results concerning density of $C_{0}^{∞}$, behaviour at infinity and integral representations for elements of the space $L^{m,p} = {⨍ | ∇^{m}⨍ ∈ L^p}$.
3
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Closed ideals in algebras of smooth functions

65%
Hanin L. G.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction........................................................................................5 1. Main definitions and basic examples..............................................7 2. Closed ideals in Sobolev algebras...............................................10  2.0. Notation...................................................................................10  2.1. Preliminary observations and results.......................................11  2.2. Closed primary ideals..............................................................13  2.3. Spectral synthesis of ideals.....................................................15 3. Spectral synthesis of ideals in the algebras $C^m Lip φ$............18 4. D-algebras...................................................................................21 5. Zygmund algebras.......................................................................26  5.1. Basic properties.......................................................................26  5.2. Extensions, approximations, and traces...................................32  5.3. Closed primary ideals...............................................................40  5.4. Point derivations......................................................................43  5.5. An extension property and spectral synthesis..........................46  5.6. Proof of Theorem 5.1...............................................................48 Appendix..........................................................................................52  1. Traces of generalized Lipschitz spaces.......................................53  2. Traces of Zygmund spaces.........................................................58  3. Proof of Proposition 5.2.11..........................................................62 References.......................................................................................65
4
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Nonlinear multivalued boundary value problems

63%
Bader R., Papageorgiou N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2001
|
tom 21
|
nr 1
127-148
EN
In this paper, we study nonlinear second order differential inclusions with a multivalued maximal monotone term and nonlinear boundary conditions. We prove existence theorems for both the convex and nonconvex problems, when $domA ≠ ℝ^{N}$ and $domA = ℝ^{N}$, with A being the maximal monotone term. Our formulation incorporates as special cases the Dirichlet, Neumann and periodic problems. Our tools come from multivalued analysis and the theory of nonlinear monotone operators.
5
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Elliptic operators on refined Sobolev scales on vector bundles

63%
Zinchenko T.
Open Mathematics
|
2017
|
tom 15
|
nr 1
907-925
EN
We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.
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