Let $\mathcal C^p$ be the space of all $p$-times continuously differentiable functions, and let $$\mathcal F = \Big\{f\in\mathcal C^p : \int\left\vert\frac{\partial^j}{\partial x^j}f(x)\right\vert^p\mathrm dx < \infty, j=0,1,2,\dots,p\Big\}$$ (i.e., the space of all $p$-th power integrable derivatives of functions in $\mathcal C^p$. Is there a certain name for the space $\mathcal F$?
$\mathcal F$ is very similar to the Sobolev space $\mathcal W^p$, except that the derivative is understood in the strong sense. Since Sobolev spaces are very important, I was wondering whether spaces like $\mathcal F$ have been studied and thus have a name?