I am trying to prove for $s>\frac{1}{2}$, a test function $f$, $$\left \| f|_{R^{d-1}} \right \|_{H^{s-\frac{1}{2}}(\mathbb{R}^{d-1})}\leq C\left \| f \right \|_{H^{s}(\mathbb{R}^{d})}$$
for some constant $C$ depending only on $s$ and the dimension $d$.
I was given a hint to use Fourier transform and Schurs lemma about integral operators but dont know how to exploit the lemma.
Under Fourier transformation, the both sides become
$$\left \| <\xi >^{s-\frac{1}{2}}\hat{f|_{\mathbb{R}^{d-1}}(\xi )} \right \|_{2} \leq C \left \| <\xi >^{s}\hat{f}(\xi) \right \|_{2}$$
and substituting the formula $\hat{f|_{\mathbb{R}^{d-1}}(\xi )}=\int_{\mathbb{R}} \hat{f}(\xi ,\eta )d\eta $,
the inequality is equivalent to
$$(\int_{\mathbb{R}^{d-1}}<\xi>^{2s-1}|\int_{\mathbb{R}}\hat{f}(\xi ,\eta )d\eta|^{2})^{1/2}\leq C(\int_{\mathbb{R}^{d}}<\xi>^{2s}|\hat{f}(\xi )|)^{1/2}d\xi)^{1/2}$$
All the above are just easy steps, but now I dont know how to use Schurs lemma.
Can I get some directions from here?
This question is from Exercise 43 in Terry Tao`s note https://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/
