Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in W^{-\alpha+1}_p(\Omega)$ if $\alpha >0$? I have found results for $\alpha=1$ and for $\alpha <0$ (i.e. the gradient is contained in a Sobolev space with positive smoothness parameter).
I would be thankfull for answers!