I am studying this paper http://143.248.27.21/mathnet/paper_file/washington/gunther/cpde.pdf and I have a doubt I have not been able to solve since yesterday. In page 14, just before the last bunch of equations, they say:
The estimate $\Vert \psi_{q} \Vert _{C(\partial \Omega)} \leq C \tau > ^{1/4},$ which follows by (0.3) and Sobolev embedding theorem...
(0.3) tells us that for any $0\leq s\leq 2,$ we have $$\Vert \psi_{q} \Vert_{H^{s}(\Omega)}\leq C\tau^{s-1}.$$
I am incapable to derive the mentioned estimate. I will share with you what I have tried, but you may stop reading from here if you already know an answer.
By Sobolev embedding, we may write $$\Vert \psi_{q} \Vert _{C(\partial \Omega)} \leq C \Vert \psi_{q} \Vert_{H^{\frac{n-1}{2}}(\partial \Omega)} $$
Now, by the trace theorem, we can bound this last quantity by:
$$C^{\prime} \Vert \psi_{q} \Vert_{H^{\frac{n+1}{2}}(\Omega)}$$
and I don't know what to do from here. Another try was to use logarithmic convexity of Sobolev norms, but if you write the details you will see that everything thus obtained is as useless...
Another question, independent of what is above I also don't know how to derive the estimate (0.3) for the Complex Geometrical Optics. I know how to do this in the construction for the problem with complete data, by using Fourier series. I am not sure whether this automatically applies here (you can see they do not prove (0.3) anywhere in the paper)
Thanks for any help you can offer