Extension theorems

1998 ◽  
pp. 247-288
Author(s):  
Victor I. Burenkov
Keyword(s):  
Extension Theorems

Friedrichs type extension theorems for multivalued operators

1991 ◽  
Vol 2 (1) ◽  
pp. 105-121
Author(s):  
George Dinca ◽  
Daniel Mateescu
Keyword(s):  
Extension Theorems ◽  
Multivalued Operators

General Extension Theorems for Linear Functionals

1974 ◽  
Vol 61 (1) ◽  
pp. 111-122 ◽  
Author(s):  
M. Landsberg ◽  
W. Schirotzek
Keyword(s):  
Linear Functionals ◽  
Extension Theorems

scholarly journals Elliptic Theory for Sets with Higher Co-dimensional Boundaries

2021 ◽  
Vol 274 (1346) ◽  
Author(s):  
G. David ◽  
J. Feneuil ◽  
S. Mayboroda
Keyword(s):  
Green Function ◽  
Elliptic Operator ◽  
Harmonic Measure ◽  
Lipschitz Constant ◽  
Extension Theorems ◽  
Content Type ◽  
Elliptic Theory ◽  
Basic Set ◽  
Degenerate Elliptic ◽  
The Green Function

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

scholarly journals Extension theorems for analytic objects associated to foliations

2012 ◽  
Vol 136 (1) ◽  
pp. 54-71 ◽  
Author(s):  
César Camacho ◽  
Bruno Scárdua
Keyword(s):  
Extension Theorems

scholarly journals Extension theorems for various weight functions over Frobenius bimodules

2018 ◽  
Vol 17 (03) ◽  
pp. 1850052 ◽  
Author(s):  
Heide Gluesing-Luerssen ◽  
Tefjol Pllaha
Keyword(s):  
Duality Theory ◽  
Extension Property ◽  
Finite Ring ◽  
Weight Functions ◽  
Hamming Weight ◽  
Theoretic Approach ◽  
Extension Theorems ◽  
Homogeneous Weight

In this paper, we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules, we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom–Tsfasman weight.

Stone–Weierstrass and extension theorems in the nonlocally convex case

2018 ◽  
Vol 462 (2) ◽  
pp. 1536-1554
Author(s):  
Vlad Timofte ◽  
Aida Timofte ◽  
Liaqat Ali Khan
Keyword(s):  
Extension Theorems ◽  
Convex Case

Extension of Monotonic Functions and Representation of Preferences

2021 ◽  
Author(s):  
Özgür Evren ◽  
Farhad Hüsseinov
Keyword(s):  
Dominance Relation ◽  
Translation Invariance ◽  
Compact Set ◽  
Continuous Real Function ◽  
Translation Invariant ◽  
Extension Theorems ◽  
Topological Vector Spaces ◽  
Dominance Relations ◽  
Universal Space ◽  
Vector Structure

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

Preliminary Material: Extension Theorems, Martingales, and Compactness

1997 ◽  
pp. 7-45 ◽  
Author(s):  
Daniel W. Stroock ◽  
S. R. Srinivasa Varadhan
Keyword(s):  
Extension Theorems
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