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Results tagged with fa.functional-analysis
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user 527557
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3
votes
2
answers
209
views
Sobolev extension problems of $W^1_\infty(\Omega)$
Recently I have read the paper Whitney's problem on extendability of functions and an intrinsic metric written by Nahum Zobin and published by Advances in Mathematics in 1998. I am confused about one …
- 69
2
votes
0
answers
133
views
Why a function induced by the infimum of the arclength of curves is Lipschitz?
Recently I have read a paper "Weighted Trudinger-type Inequalities" written by Stephen M. Buckley and Julann O'Shea and published by Indiana University Mathematics Journal in 1999, MR1722194, Zbl 0965 …
- 69
0
votes
0
answers
141
views
A Poincaré inequality holds for $p>2$ but fails for $p\leqslant 2$
I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).
Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset …
- 69
0
votes
0
answers
105
views
A Lipschitz function induced by the infimum of the length of curves
Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of t …
- 69
0
votes
0
answers
94
views
Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with t...
A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if
\begin{equation*}
\dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K,
\end{equation*}
for any $x,y\in \ma …
- 69