For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.

Sobolev spaces are function spaces generalizing the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, derivatives of functions in a Sobolev space are also required to be sufficiently integrable: that is, we require all (weak) partial derivatives of the function up to a certain order belong to a certain fixed Lebesgue space.

In more detail, let $U \subseteq \mathbb{R}^n$ be an open set. A weak $\alpha$th partial derivative $D^\alpha f$ of $f$ is a function $g\in L^1_{\mathrm{loc}}(U)$ such that $$\int_U f D^\alpha \phi \, dx = (-1)^{|\alpha|} \int g\phi \, dx$$ for each compactly supported smooth function $\phi \in C^\infty_c(U)$. the Sobolev space $W^{k, p}(U)$ consists of those functions $f\in L^p(U)$ such that for every multi-index $\alpha$ of length at most $k$, every weak partial derivative $D^{\alpha}f$ exists and is an element of $L^p$. The Sobolev spaces are equipped with norms defined by

$$\|u\|_{W^{k,p}(U)} = \begin{cases} \left( \sum_{|\alpha| \le k} \|D^{\alpha} u\|_{L^p(U)}^p \right)^{1/p} & p < \infty, \\ \max_{|\alpha| \le k} \|D^{\alpha}\|_{L^{\infty}(U)} &p=\infty .\end{cases}$$

$(W^{k,p}(U),\|\cdot\|_{W^{k,p}(U)})$ are Banach spaces for each $k\in\mathbb N$, and each $p\in[1,\infty]$. The norms measure both the size and the regularity of a function.

The basic fundamental result of Sobolev spaces is the Sobolev embedding theorem. In words, it says that (1) if $kp<n$, having $k$ weak derivatives in $L^p$ places your function in a better Lebesgue space $L^{p^*}$, where $\frac1{p^*} = \frac1p - \frac kn$, and (2) if $kp>n$, then your function is not only in $L^\infty$ but also has a continuous representative in some space of Hölder continuous functions. In particular, sufficiently many weak derivatives means that your function is in fact classically differentiable.

Reference: Sobolev space.