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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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 …
Javier's user avatar
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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 …
Javier's user avatar
  • 69
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 …
Javier's user avatar
  • 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 …
Javier's user avatar
  • 69
0 votes
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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 …
Javier's user avatar
  • 69