Definition of Sobolev space via distribution

Question

Can someone give me some comments about the following definition of the Sobolev space $W^{k,p}(U)$ for $U \subset \mathbb{R}^d$ open, $k\in \mathbb{N}$ and $1 \leq p \leq \infty$? The Sobolev space is the set of all $L^p (U)$ functions, such that for all multi indices $\alpha$ with $\vert\alpha\vert \leq k$ there exists $f_\alpha \in L^p (U)$ so that $\partial^\alpha T_f = T_{f_\alpha}$ (in the sense of distributions). What is the weak derivative of what? I am completely overextended... I just need some lines about this definition. That would help a lot.

Answer 1

Every function $f: U\to \mathbb R$ which is locally $L^1$ (i.e., $\int_K |f(x)|dx <\infty$ for every compact set $K$) induces a distribution $T_f: \mathscr D(U)\to\mathbb R$, $\varphi \mapsto \int \varphi(x)f(x)dx$ (which is correctly defined because $\varphi$ has compact support). The distributional derivative $\partial^\alpha T$ is defined as $\varphi\mapsto (-1)^{|\alpha|}T(\partial^\alpha \varphi)$. The requirement in the definition of Sobolev spaces is that these distributional derivatives of $T_f$ are again induced by $L^p$-functions.