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== 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 measurable 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}