# Contents

## Idea

If $1\le p<\infty$ and $\Omega$ is a domain (in a $n$-dimensional real space with easy generalization to manifolds), one first considers the Lebesgue spaces ${L}_{p}={L}_{p}\left(\Omega \right)$ (wikipedia) of (equivalence classes of) measurable (complex- or real-valued) functions $f$ whose (absolute values of) $p$-th powers are Lebesgue integrable; i.e. whose norm

$\parallel f{\parallel }_{{L}_{p}}={\left({\int }_{\Omega }\mid f{\mid }^{p}d\mu \right)}^{1/p}$\| f\|_{L_p} = \left(\int_\Omega |f|^p d\mu\right)^{1/p}

is finite. For $p=\infty$, one looks at the essential supremum norm $\parallel f{\parallel }_{{L}_{\infty }}$ instead.

For $1\le p\le \infty$, and $k\ge 1$ the Sobolev space ${W}_{p}^{k}={W}_{p}^{k}\left(\Omega \right)$ or ${W}^{k,p}\left(\Omega \right)$ is the space of measurable functions $f$ on $\Omega$ such that its generalized partial derivatives ${\partial }_{1}^{{i}_{1}}\dots {\partial }_{n}^{{i}_{n}}f$ (e.g. in the sense of generalized functions) for all multiindices $i=\left({i}_{1},\dots ,{i}_{n}\right)\in {ℤ}_{\ge 0}^{n}$ with ${i}_{1}+\dots +{i}_{n}\le k$ are in ${L}_{p}\left(\Omega \right)$. The most important case is the case of the Sobolev spaces ${H}^{k}\left(\Omega \right):={W}_{2}^{k}\left(\Omega \right)$. Sobolev spaces are particularly important in the theory of partial differential equations.

## References

• L. C. Evans, Partial Differential Equations, Amer. Math. Soc. 1998.