User talk:Nikita2

Weighted Sobolev Spaces

Let $D\subset \mathbb R^n$ be open and let $w:\mathbb R^n\rightarrow[0,\infty)$ be a locally summable nonnegative function "weight". For $1\leqslant p<\infty$ and $l\in\mathbb N$ we can define weighted Sobolev space $W^l_p(D,w)$ as the set of locally summable functions $f:D\to\mathbb R$ such that for every multi-index $\alpha$ there exists weak derivative $D^{\alpha}f$ and

\begin{equation} \|f\mid W^l_p(D, w)\| = \Biggl(\,\sum\limits_{|\alpha|\leqslant l}\ \int\limits_{D}|D^{\alpha}f|^p(x)w(x)\, dx \,\Biggr)^{\frac{1}{p}} < \infty. \end{equation}

One of conjectures of De Giorgi

If $\exp(tw)$, $\exp(tw^{-1}) \in L^1_{\operatorname{loc}}$ for each $t > 0$ then $w$ is regular weight.

Welcome

Hello Nikita2 and welcome! Are you also a descendant of De Giorgi?

I saw you texxed the page Luzin-C-property. I was thinking some time ago to start renaming all pages using Lusin and set automatic redirections for the ones with Luzin. What do you think about it? Camillo (talk) 12:10, 25 November 2012 (CET)

Actually I am not (yet) a descendant of De Giorgi, I am trying to solve pair of these conjectures.

I think there should be definitions in more general form: if a measurable mapping $f:X\to Y$, $|X|<\infty$ then for any $\varepsilon$ there is an open set $B$ such that the function $f$ is continuous on $B$ and $|X \setminus B|<\varepsilon$ The same for N-property. I am going to do it soon. --Nikita2 (talk) 09:56, 26 November 2012 (CET)

Yes, there is a lot to do (for instance on Sobolev spaces there are plenty of stuff missing: I could not find any place where Poincare' and Sobolev inequalities are mentioned!). I am glad you joined us. On my userpage you can see what I have been doing and also a tentative list of pages to create and to update. Another user who is looking at these topics is User:Matteo.focardi: he updated the page on Egorov's theorem. Camillo (talk) 10:30, 26 November 2012 (CET)