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In Theorem 5.2 of the book

Robert A. Adams and John J. F. Fournier, MR 2424078 Sobolev spaces, ISBN: 0-12-044143-8.

is stated that, for any integers $0 \leq j \leq m$ and $u \in W^{m,p}(\Omega)$ (under suitable assumptions on $\Omega$), the following interpolation inequality holds $$ \|u\|_{W^{j,p}} \lesssim \|u\|^{j/m}_{W^{m,p}} \cdot \|u\|_{L^p}^{(m-j)/m}. $$

Is there an analogous result for Sobolev spaces of fractional order? (I am particularly interested in the case $p=2$).

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  • $\begingroup$ Inequalities like this are pretty much the point of interpolation spaces. There are whole books about the subject. $\endgroup$
    – Michael Renardy
    Jul 18, 2016 at 13:10
  • $\begingroup$ Thank you for the comment. Could you please suggest me a good reference? $\endgroup$
    – Paglia
    Jul 18, 2016 at 13:37

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