Proof of existence of trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ not using the Fourier transform

Question

I'm looking for a proof of the existence of the trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ which does not use the Fourier transform.

In particular, I want to prove the following inequality without using Fourier methods $$|u(x,0)|_{H^{\frac 12}} \leq C\lVert{u}\rVert_{H^1}$$ for appropriately smooth functions which then can be extended by density.

Does anyone have a source for this? Thanks.