For $p<0$, we defined the semi-normed space $L^p(\Omega )$ where $\Omega \subset \mathbb R^n$ as following :
Set $s=\left\lfloor \frac{-n}{p}\right\rfloor $ and $-\alpha =s+\frac{n}{p}$, and we define the norm $\|\cdot \|_p$ as following :
$$\|u\|_p=\begin{cases}\sup \{[D^s u]_\alpha^{\Omega }&\alpha >0\\ \sup |D^s u|&\alpha =0 \end{cases},$$ where the supremum is taken over all partial derivatives $D^s$ of order $s$, $[f]_\alpha ^\Omega =\sup_{x,y\in \Omega }\frac{|f(x)-f(y)|}{|x-y|^\alpha }$, and for $(k_1,...,k_n)\in \mathbb N$ s.t. $k_1+...+k_n=s$, we have that $D^s=D_1^{k_1}...D_n^{k_n}$.
So function are in $L^p$ if $\|u\|_p<\infty $.
How can we give a sense to this definition, and what is the motivation for such spaces ?