Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that $$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$ where $u|_{\partial\Omega}=\phi$.

Can we estimate $L^r$-norm of $u$ for higher $r$ in terms of some norm in $\phi$ ? More specifically I am interested to know if $\|u\|_{L^{2(p-1)}}$ is bounded by some $L^r(\partial\Omega)$-norm of $\phi$?

Any help or reference will be greatly appreciated.