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.