# A inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator?

Does someone know any inequality between $$||u||_{p}$$ and $$||\gamma (u)||_{p, \partial \Omega}$$, where $$\gamma$$ is the Trace Operator? I need to find something like

## $$||u||_{p}\leq C||\gamma (u)||_{p, \partial \Omega}$$ or $$||u||_{p}\leq C(||\gamma (u)||_{p, \partial \Omega} + ||\nabla u||_{p})$$ ,

but I couldn't find any in the bibliography.

Here $$||u||_{p}=||u||_{L^{p}}$$, $$\Omega$$ Lipschitz's Domain.
Not a proof, but an idea too long for comments that could maybe become a proof...I think you can let $$\phi_n\ge 0$$ be a smooth function with $$\phi_n = 1$$ on $$\partial \Omega$$ and support in $$\{x\in\Omega : d(x,\Omega) < 1/n\}$$ with $$|\nabla \phi_n| \sim Cn^\alpha d(x,\Omega)$$ there, for some constants $$C,\alpha$$ I don't know yet. Then by Poincare for functions in $$W^{1,p}_0(\Omega)$$, $$\|u\|_p \le \| u \phi_n \|_p + \|u(1-\phi_n)\|_p \lesssim \| u \phi_n \|_p + \|\nabla u(1-\phi_n)\|_p + \|u\nabla\phi_n\|_p$$ $$\| u \phi_n \|_p \to 0$$, $$\|\nabla u (1-\phi_n)\|_p \to \|\nabla u\|_p$$, and I'm guessing that for the correct choice of $$C,\alpha$$ that might depend on $$p$$, $$\|u\nabla \phi_n \|_p \to \|\gamma (u)\|_{L^p(\partial \Omega)}$$ with the hope that $$\nabla \phi_n$$ "converges in some sense" to the surface measure on $$\partial \Omega$$?