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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!

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  • $\begingroup$ Are $a$ and $\alpha$ related ? $\endgroup$ – user37238 Mar 5 '14 at 10:25
  • $\begingroup$ Yes, they are the same :) I've replaced it $\endgroup$ – Framl Mar 5 '14 at 10:45
  • $\begingroup$ $L^p(\Omega) \hookrightarrow W^{-\alpha,p}(\Omega)$ for all $\alpha>0$. $\endgroup$ – Tomás Mar 5 '14 at 20:03
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    $\begingroup$ The answer essentially depends on definitions of the spaces you are emlpoying. For $\alpha<0$, your problem is solved by the Sobolev representation theorem of a weakly differentiable function in terms of its gradient. For $\alpha>0$, it is not the gradient, it is its conjugate -- the divergence operator that solves your problem, which is much more complicated when homogeneous boundary data is involved. $\endgroup$ – mkl314 Mar 10 '14 at 11:27
  • $\begingroup$ The space $W^\alpha(\Omega)$ for negative $\alpha$ is defined as the dual space of $W^{-\alpha,0}_{p'}(\Omega)$ with $1/p+1/p'=1$. So $W^{-\alpha,0}_{p'}(\Omega)$ is the closure of the set of testfunctions with respecht to the Sobolev norm. $\endgroup$ – Framl Mar 10 '14 at 11:30

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