The Whitney problem of existence of a linear extension operator

1997 ◽  
Vol 7 (4) ◽  
pp. 515-574 ◽  
Author(s):  
Yuri Brudnyi ◽  
Pavel Shvartsman
Keyword(s):  
Linear Extension ◽  
Extension Operator ◽  
Linear Extension Operator

scholarly journals A linear extension operator for Whitney fields on closed o-minimal sets

2008 ◽  
Vol 58 (2) ◽  
pp. 383-404
Author(s):  
Wiesław Pawłucki
Keyword(s):  
Linear Extension ◽  
Extension Operator ◽  
Minimal Sets ◽  
Linear Extension Operator

scholarly journals Extension operators for Sobolev spaces commuting with a given transform

1998 ◽  
Vol 40 (2) ◽  
pp. 291-296 ◽  
Author(s):  
Viktor Burenkov ◽  
Bert-Wolfgang Schulze ◽  
Nikolai N. Tarkhanov
Keyword(s):  
Real Axis ◽  
Sobolev Spaces ◽  
Linear Extension ◽  
Extension Operator ◽  
Bounded Linear ◽  
The Real ◽  
Linear Extension Operator

AbstractWe consider a real-valued function r = M(t) on the real axis, such that M(t) < 0 for t < 0. Under appropriate assumptions on M, the pull-back operator M* gives rise to a transform of Sobolev spaces Ws.p (-∞, 0) that restricts to a transform of Ws.p(-∞, ∞). We construct a bounded linear extension operator Ws.p(-∞, 0) → Ws.p(−∞, ∞), commuting with this transform.

scholarly journals A bounded linear extension operator for L^(2,p)(R^2)

2013 ◽  
Vol 178 (1) ◽  
pp. 183-230 ◽  
Author(s):  
Arie Israel
Keyword(s):  
Linear Extension ◽  
Extension Operator ◽  
Bounded Linear ◽  
Linear Extension Operator

WHITNEY'S TYPE THEOREMS FOR INFINITE DIMENSIONAL SPACES

2000 ◽  
Vol 03 (01) ◽  
pp. 141-160
Author(s):  
S. A. SHKARIN
Keyword(s):  
Linear Extension ◽  
Linear Subspace ◽  
Extension Operator ◽  
Fréchet Space ◽  
Frechet Space ◽  
Entire Space ◽  
Closed Linear Subspace ◽  
Infinite Dimensional ◽  
Class C ◽  
Linear Extension Operator

It is proved that for any f ∈ Ck(L,ℝ), where k ∈ ℕ and L is a closed linear subspace of a nuclear Frechét space X, the function f can be extended to a function of class Ck-1 defined on the entire space X. It is also proved that for any f ∈ Ck (L, ℝ), where k ∈ℕ∪{∞} and L is a closed linear subspace of a conjugate X of a nuclear Frechét space, the function f can be extended to a function of class Ck defined on the entire space X. In addition, it is proved that under these conditions, the existence of a linear extension operator is equivalent to the complementability of the subspace.

scholarly journals OPERATORS THAT ARE NUCLEAR WHENEVER THEY ARE NUCLEAR FOR A LARGER RANGE SPACE

2004 ◽  
Vol 47 (3) ◽  
pp. 679-694 ◽  
Author(s):  
Eve Oja
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Linear Extension ◽  
Approximation Property ◽  
Closed Subspace ◽  
Extension Operator ◽  
Range Space ◽  
Bounded Linear ◽  
Linear Extension Operator

AbstractLet $X$ be a Banach space and let $Y$ be a closed subspace of a Banach space $Z$. The following theorem is proved. Assume that $X^*$ or $Z^*$ has the approximation property. If there exists a bounded linear extension operator from $Y^*$ to $Z^*$, then any bounded linear operator $T:X\rightarrow Y$ is nuclear whenever $T$ is nuclear from $X$ to $Z$. The particular case of the theorem with $Z=Y^{**}$ is due to Grothendieck and Oja and Reinov. Numerous applications are presented. For instance, it is shown that a bounded linear operator $T$ from an arbitrary Banach space $X$ to an $\mathcal{L}_\infty$-space $Y$ is nuclear whenever $T$ is nuclear from $X$ to some Banach space $Z$ containing $Y$ as a subspace.AMS 2000 Mathematics subject classification: Primary 46B20; 46B28; 47B10

Topics in geometric analysis and harmonic analysis on spaces of homogeneous type

2015 ◽  
Author(s):  
◽  
Ryan Alvarado
Keyword(s):  
Geometric Analysis ◽  
Extension Operator ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
University Of Missouri ◽  
Singular Drift ◽  
Linear Extension Operator ◽  
The University ◽  
Extension Result ◽  
The Relationship

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The present dissertation consists of three main parts. One theme underscoring the work carried out in this dissertation concerns the relationship between analysis and geometry. As a first illustration of the interplay between these two branches of mathematics we develop a sharp theory of Hardy spaces in the setting of spaces of homogeneous type. The presented work is in collaboration with M. Mitrea. In the second part, we prove that a function defined on a subset of a geometrically doubling quasi-metric space which satisfies a Holder-type condition may be extended to the entire space with preservation of regularity. The proof proceeds along the lines of the original work of Whitney in 1934 and yields a linear extension operator. A similar extension result is also proved in the absence of the geometrically doubling hypothesis, albeit the resulting extension procedure is nonlinear in this case. This work is done in collaboration I. Mitrea and M. Mitrea. In the final part of the dissertation we prove that an open, proper, nonempty subset of Rn is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. Additionally, we prove a sharp generalization of the Hopf-Oleinik boundary point principle for domains satisfying a one-sided, interior pseudo-ball condition, for semi-elliptic operators with singular drift. These results have been obtained in collaboration with D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziade.

A Continuous Extension Operator for Convex Metrics

2010 ◽  
Vol 53 (4) ◽  
pp. 719-729
Author(s):  
I. Stasyuk ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Extension ◽  
Extension Operator ◽  
Uniform Topology ◽  
Simultaneous Extension ◽  
Peano Continuum ◽  
Convex Metrics

AbstractWe consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.

Calculating direct and indirect contributions of players in cooperative games via the multi-linear extension

2018 ◽  
Vol 164 ◽  
pp. 27-30
Author(s):  
André Casajus ◽  
Frank Huettner
Keyword(s):  
Cooperative Games ◽  
Linear Extension

EXPONENTIAL MEAN SQUARE STABILITY OF STOCHASTICALLY FORCED INVARIANT MANIFOLDS FOR NONLINEAR SDEs

2007 ◽  
Vol 07 (03) ◽  
pp. 389-401 ◽  
Author(s):  
L. B. RYASHKO
Keyword(s):  
Linear Extension ◽  
Invariant Manifolds ◽  
Function Technique ◽  
Nonlinear Stochastic System ◽  
General Notion ◽  
Mean Square ◽  
Extension System ◽  
Mean Square Stability ◽  
The Stability ◽  
Exponential Mean Square Stability

An exponential mean square stability for the invariant manifold [Formula: see text] of a nonlinear stochastic system is considered. The stability analysis is based on the [Formula: see text]-quadratic Lyapunov function technique. The local dynamics of the nonlinear system near manifold is described by the stochastic linear extension system. We propose a general notion of the projective stability (P-stability) and prove the following theorem. The smooth compact manifold [Formula: see text] is exponentially mean square stable if and only if the corresponding stochastic linear extension system is P-stable.

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