So say I have $f\in L^p$. Is there always a function $g\in L^p$ such that $$ \langle f,-\phi_x\rangle = \langle g,\phi\rangle $$ for all $\phi\in C^\infty_c$. Or phrased differently if $f\in L^p$ does $g\in W^{1,p}$ with $g_x=f$ exist?
(at least on bounded domains)
I think so but I'm not sure. My proof would be:
Define the distribution $T_{g(f)}(\phi) = \langle f,-\phi_x\rangle$. Then in a lecture I was told that if $T$ is a distribution and $\partial_1 T,\dots,\partial_n T\in \text{RM}_{\text{loc}}$ then $T\in L^1_\text{loc}$. So by Hölder's inequality $\partial_x T_g\in L^1_{\text{loc}}$ such that $T_g\in L^1_\text{loc}$. Using the same technique again I get that the second antiderivative is $L^1_{\text{loc}}$ and therefore in $W^{1,1}_{\text{loc}}$ which in turn by Sobolev embedding implies that $T_g$ is continuous and therefore $L^p_{\text{loc}}$
Is this true?