Relation between fractional and integer Sobolev norms

Question

I encountered a situation where I have to add two norms defined on the boundary:

$$C_1 ||u||^2_{L_2(\partial\Omega)} + C_2 ||u||^2_{H^{3/2}(\partial \Omega)},$$

but do not really know how to manipulate this expression. Is there a relation between the norms?

Answer 1

1. There is a trivial relation $\|u\|_{L_2}\leq\|u\|_{H^1}$
2. The Sobolev embedding theorem as well as other similar results may give you some tools to use to relate Sobolev norms to $L_p$ norms and other Sobolev norms. See here or chapter 5 of Evans for more details.
3. The generalized Hölder inequality (see here) may also prove useful if you can bound the Sobolev norm in terms of some $L_p$ norm. Assuming your boundary is compact, you can take the 2 functions in the inequality to be $1$ and $u$, obtaining bounds between different $L_p$ norms with a constant depending on $\partial\Omega$.

Without knowing more about what you are trying to show it's hard to know exactly which sort of inequalities you need.