Let $\Omega\subseteq\mathbb{R}^{n}$ be an open subset and $k\in\mathbb{N}$. We write $C^{k}(\Omega)$ for the space of real valued, $k$ times continuously differentiable functions on $\Omega$. Furthermore we denote by $L_{loc}^{1}(\Omega)$ the space of locally integrable functions on $\Omega$ i.e. functions $f:\Omega\rightarrow\mathbb{R}$ such that the restriction of $f$ to any compact subset of $\Omega$ is integrable. Let $f,g\in L_{loc}^{1}(\Omega)$ and $\beta\in\mathbb{N}^{n}$. We say that $g$ is the $\beta$-th weak derivative of $f$ if for any test function $\varphi\in C^{\infty}(\Omega)$ with compact support we have

$$\int_{\Omega}fD^{\beta}\varphi\;\boldsymbol{dx}=(-1)^{|\beta|}\int_{\Omega}g\varphi\;\boldsymbol{dx}$$

where $\boldsymbol{dx}$ denotes the Lebesgue measure on $\mathbb{R}^{n}$. We denote by $W^{k,1}_{loc}(\Omega)$ the space of locally integrable functions on $\Omega$ whose weak derivatives exist up to order $k$.