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(optimal) Constant in Sobolev extension Theorems

It is well known that for Lipschitz domains $\Omega\subset \Bbb{R}^2$ one can define linear extension operators $$ E: H^1(\Omega) \to H^1(\Bbb{R}^2).$$ I am interested in explicit upper bounds for the ...
Beni Bogosel's user avatar
  • 2,172
1 vote
1 answer
96 views

A correspondence between projective representations of $G$ with those of its universal cover

Let $G$ be a connected Lie group and $\mathcal{H}$ be a Hilbert space. Let $U(\mathcal{H})$ denote the the group of all unitary operators on $\mathcal{H}$ with function composition (i.e., $\hat{U}:\...
Mahtab's user avatar
  • 277
2 votes
2 answers
104 views

Extensions of $G$-modules parametrized by $H^1$

Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess one can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
Conjecture's user avatar
1 vote
0 answers
59 views

How can we construct a non-trivial central extension of a Lie group

Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
Mahtab's user avatar
  • 277
4 votes
1 answer
171 views

Explicit formula for general group extension in terms of cartesian product set

According to Wikipedia and ncat lab general group extensions $$N\rightarrow G\rightarrow Q$$ are classified by a group homomorphism $$\rho: Q\rightarrow \operatorname{Out}(N)$$ subject to a constraint ...
Andi Bauer's user avatar
  • 2,911
5 votes
2 answers
438 views

How is the classification of groups extensions by $H^2$ related to Yoneda Ext?

It is well-known that group extensions $$1\to A \to H \to G \to 1$$ where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...
Antoine Labelle's user avatar
0 votes
1 answer
182 views

Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$

Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...
Andrea Aveni's user avatar
2 votes
0 answers
137 views

Homotopy equivalence of chain complexes from subcomplexes and quotient complexes

Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
Faniel's user avatar
  • 653
5 votes
1 answer
174 views

On the property P in the Whitney extension theorem

Let $D$ be a possibly unbounded domain in $\mathbb{R}^d$, $d \ge 2.$ We say that $D$ has the property P if there exists $C>0$ such that such that any pair of points $x,y \in D$ can be joined by a ...
sharpe's user avatar
  • 701
1 vote
0 answers
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extension from a dense subset in completely uniformizable spaces

Let $\mathbf{CReg}$ the category of completely regular spaces and continuous maps and let $\mathbf{Unif}$ be the category of uniform spaces and uniformly continuous maps. There is a functor $F:\mathbf{...
Ruben Van Belle's user avatar
3 votes
1 answer
176 views

Homeomorphic extension of a discrete function

Let $f : \{ 0,1 \} ^ {n} \rightarrow \{ 0,1 \} ^ {n}$ be a bijective map. Then is there a known computable way to extend it to a homeomorphism $g:[ 0,1 ] ^ {n} \rightarrow [ 0,1 ] ^ {n}?$
Guill Guill's user avatar
2 votes
1 answer
170 views

Lie algebras for which all one-dimensional extensions split

I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will ...
Nikhil Sahoo's user avatar
  • 1,205
4 votes
2 answers
153 views

A ball with slit at the radius is not $W^{1,1}$-extension domain

Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
Guy Fsone's user avatar
  • 1,043
3 votes
1 answer
151 views

Boundedness of an extension operator

Let $d \ge 2$ be a positive integer. For $x=(x_1,\dotsc,x_{d-1},x_d)$, we write $x'=(x_1,\dotsc,x_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x_d) \mid x_d>0\}$ denote the $d$-dimensional upper half-space. ...
sharpe's user avatar
  • 701
2 votes
0 answers
109 views

Any connection between extension of algebraic structure and forcing of set theory?

Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
XL _At_Here_There's user avatar
5 votes
1 answer
269 views

Extension of first order deformations of a line bundle

Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
Javier Gargiulo's user avatar
1 vote
0 answers
46 views

$H^1 \cap C^0$ boundary, smooth $H^1$ extension

Assume we have a $u \in H^1(\Omega; \mathbb{R}^n) \cap C^0$ where $\Omega$ is a bounded open Set with smooth boundary. Also $u\vert_{\partial \Omega} \in H^1(\partial \Omega; \mathbb{R}^n) \cap C^0$. ...
Kilian Koch's user avatar
2 votes
0 answers
82 views

Reference for an extension theorem for Neumann boundary data

$\DeclareMathOperator\Tr{Tr}$Let $\Omega \subset \mathbb{R}^d$ be a smooth bounded domain (we denote by $n$ the normal to $\partial\Omega$) and $p\in(1,\infty)$. Do you know where I can find (book or ...
J.Mayol's user avatar
  • 489
8 votes
3 answers
726 views

Is there some example that nicely extends the multiplication of natural numbers?

Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The ...
Zelox's user avatar
  • 181
3 votes
1 answer
526 views

Yoneda Ext theorem and extensions

Consider the category of chain complexes over a ring $R$. We can show that $\text{Ext}^1(M, N)$ classifies extensions using the triangulated category structure: the homotopy kernel of a map $N \...
user avatar
2 votes
1 answer
408 views

$C^1$ extension with compact support

Knowing that $\omega\Subset\Omega\subset\mathbb{R}^2$ (compactly included) are two open and bounded sets with $C^2$ boundary, is it true that for any function $\phi_0:\overline{\omega}\to\mathbb{R},\ \...
Bogdan's user avatar
  • 1,414
2 votes
0 answers
112 views

Extensions in a full subcategory

Let $\mathcal{C}$ be an abelian category (feel free to put more adjectives here) and $\mathcal{D}$ a full abelian subcategory closed under kernels and cokernels. Then by definition for $A,B\in \...
user197402's user avatar
4 votes
0 answers
145 views

Continuous extension preserving modulus of continuity

Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
Catologist_who_flies_on_Monday's user avatar
2 votes
0 answers
263 views

Extension of Lipschitz functions that preserve the Frobenius norm of the Jacobian

Let $n,m\ge 1$ be integers and let $f:E\to R^m$ be $L$-Lipschitz for some subset $E\subset R^n$. Kirszbraun's theorem, https://en.wikipedia.org/wiki/Kirszbraun_theorem, states that there exists ...
jlewk's user avatar
  • 1,564
2 votes
0 answers
56 views

Extension of differentiable structure to guarantee continuous extension of prescribed vector fields

Consider the lower half space $\{(x,y) \in \mathbb{R}^2 \; | \; y < 0\}$ and let $X_1, X_2$ be two vector fields on the lower half space which are continuous with respect to its canonical ...
jsb's user avatar
  • 383
6 votes
0 answers
234 views

How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
DJWilliams's user avatar
1 vote
0 answers
48 views

Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?

Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$. Let $...
kaka Hae's user avatar
  • 117
6 votes
1 answer
161 views

Extension Operator for $W^{1,\infty}(U,X)$

I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
Sibyl Osullivan's user avatar
1 vote
1 answer
136 views

Computation of extension groups in the category of pairs $(M,f)$

Let $A$ be a unitary commutative ring, and let $B$ be an $A$-algebra. We consider the category whose objects are pairs $\textbf{M}=(M,f)$ where $M$ is an $A$-module and where $f$ is a $B$-linear ...
Stabilo's user avatar
  • 1,479
2 votes
1 answer
212 views

Homomorphisms of ring extending nicely ideal intersections

Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$. Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$. Is there some "natural" assumption on $\varphi$ to ...
MonLau's user avatar
  • 43
1 vote
1 answer
107 views

Extend a bundle "trivially"

Suppose I have a fibre bundle $E\to B$ with compact fibre. Furthermore, $B$ is open in a larger, compact space, e. g. $B\subseteq B'$. I want to get a map $E'\to B'$ (not a bundle any more!) with $E'|...
FKranhold's user avatar
  • 1,623
21 votes
1 answer
752 views

Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure

Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-...
M. Winter's user avatar
  • 12.8k
1 vote
0 answers
154 views

Finite field extension

Suppose $$f_a=x^p-x-\left [\prod_{j=1}^{a-1} \alpha_j\right]^{p-1} \in \mathbb{F}_p(\alpha_1,\ldots,\alpha_{a-1})[x]$$ is irreducible over $\mathbb{F}_p(\alpha_1,\ldots,\alpha_{a-1}) $, where $\...
Jannes Braet's user avatar
4 votes
0 answers
253 views

Do all fields with internal absolute values arise as ordered fields or like $\mathbb{C}$ from them?

$\def\abs#1{\lvert#1\rvert} \def\Im{\operatorname{Im}} \def\Re{\operatorname{Re}}$ (Crossposted from math.stackexchange.com after 5 days with no correct answer.) Let ​ $\langle F,+,\cdot\rangle$ ​ be ...
user avatar
5 votes
2 answers
2k views

Extensions of local vector fields to whole manifold

Let $M$ be a smooth manifold (with boundary). Suppose I have a smooth vector field $T$ defined on the complement of a compact subset $K$ of $M$ and I wish to extend $T$ to the whole of $M$. What are ...
onamoonlessnight's user avatar
2 votes
0 answers
67 views

Ref Request: Extension Operators for Slobodeckii Spaces of Higher Order

I have been looking for (linear) Extension Operators for Slobodeckii spaces $W^{s,p}(\Omega)$ where $s>1$ and $\Omega \subset\mathbb{R}^N$ is a sufficiently smooth domain, where the influence of $\...
deMiranda's user avatar
  • 351
3 votes
1 answer
409 views

busby invariant of extensions of $C^*$-algebras

I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras". Let $0\to B\to E\to A\to 0$ be a short exact ...
user avatar
1 vote
0 answers
448 views

Quotient of two smooth functions extension

Assume we are given smooth functions $f, g: U \to \mathbb{C}$, where $0 \in U \subset \mathbb{R}^n$ is open and $0 \in g^{-1}(0) \subset \{x_n = 0\}$. Furthermore, suppose that $\nabla g \neq 0$ on ...
Ceka's user avatar
  • 501
5 votes
1 answer
350 views

Extension of functions from geodesically convex compact sets in a Riemannian manifold

In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
David Roberts's user avatar
  • 34.8k
9 votes
0 answers
113 views

General approaches to extension theorems as Caratheodory

I would like to know if there are some general studies about extension-like theorem, in the sense which i'm going to describe. This paragraph is not rigorous; I just would like the idea to be clear. I ...
Andrea Marino's user avatar
3 votes
1 answer
3k views

Extension of continuous and smooth functions

Let us consider any subset $U \subset \mathbb{R}^{n}$. By definition, a function $f: U \rightarrow \mathbb{R}^m$ is smooth if, for every $x \in U$, there exist an open neighbourhood $\Omega_{x}$ of $x$...
Fabio's user avatar
  • 1,222
3 votes
1 answer
869 views

Fractional Sobolev spaces and extension by zero

The all-wise Wikipedia suggest (https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero) that for $s > 1/2$ any function from $H^s_0(\Omega)$ can be extended by zero to whole $\mathbb{R}^n$ (...
user78417's user avatar
5 votes
2 answers
453 views

Conjugation in associative algebras over finite fields

Let $A$ be a finite dimensional associative algebra (with unity) over a finite field $F$. Let $L$ be a field extension of $F$. Suppose that after extending scalars to $L$, two elements $a,b$ of $A$ ...
Oliver's user avatar
  • 367
3 votes
1 answer
2k views

Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?

A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...
NeverConvex's user avatar
-1 votes
1 answer
207 views

Ext functor for more than two modules? [closed]

The question is natural. Let's just work in the category of modules over a ring. Pick three modules $M_1, M_2, M_3$. Consider consecutive extensions of these modules, i.e., consider M, such that we ...
tqvb's user avatar
  • 11
4 votes
1 answer
305 views

Triviality of local system extension

Take a nice space $X$. Let us call a local system on $X$ a functor from the fundamental groupoid of $X$ to groups, so that $G$ is a local system on $X$ if for each $x \in X$ there is a group $G_x$ and ...
Frol Zapolsky's user avatar
17 votes
2 answers
2k views

The letters of the word "ART"

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is $...
Ali Taghavi's user avatar
4 votes
0 answers
146 views

A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor. Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of ...
Ali Taghavi's user avatar
2 votes
1 answer
375 views

Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$. As one could find in G.M. Troianello "Elliptic Differential Equations and ...
foo90's user avatar
  • 291
0 votes
1 answer
154 views

Units of an extension of $\mathbb{Z}$ [closed]

Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there ...
Mayank Pandey's user avatar