# What exactly is the "real interpolation space" $(L^q, W^{2,q})_{1-p^{-1},p}$ for $1<p,q<\infty$?

I came across the notation https://en.wikipedia.org/wiki/Interpolation_space#Real_interpolation the real interpolation space is discussed for Bessel potential spaces.

However, I do not see exactly how to apply this general theory to the specific case $$$$(L^q(\Omega), W^{2,q}(\Omega))_{1-\frac{1}{p},p}$$$$ for $$1 and a bounded region $$\Omega$$ in $$\mathbb{R}^n$$ with smooth boundaries.

Could anyone pleaes clarify for this case? In particular, do I have to use Bessel potential spaces or Sobolev–Slobodeckij spaces?

Identifying interpolation spaces is in general a non-trivial task. In your case you get certain Besov spaces; it is shown in Theorem 17.24 of

Leoni, Giovanni, A first course in Sobolev spaces, Graduate Studies in Mathematics 181. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2921-8/hbk; 978-1-4704-4226-2/ebook). xxii, 734 p. (2017). ZBL1382.46001.

that we have the identification $$(L^q(\mathbb R^n), W^{k,q}(\mathbb R^n))_{\sigma,p} = B^{\sigma k,q}_p(\mathbb R^n).$$ So in your case, we would expect $$B^{2(1-\frac1p),q}_p(\Omega)$$, which is indeed what you get. Besov spaces on bounded (and regular) domains can be defined in several equivalent ways, such as the restriction of functions on $$B^{s,p}_q(\mathbb R^n)$$ to $$\Omega$$. See for instance the discussion at the end of Section 17.3 of the aformentioned text.

Other characterisations, and a more comprehensive treatment of these spaces can be found in Chapter I.3 of

Triebel, Hans, Theory of function spaces, Monographs in Mathematics, Vol. 78. Basel-Boston-Stuttgart: Birkhäuser Verlag, DM 90.00 (1983). ZBL0546.46027.

• Thank you for your answer. I have been looking into Besov spaces as well. Is it true in general that $W^{k,q}$ is continuously embedded in the Besov space you wrote above? Commented Aug 27, 2023 at 15:09
• @Keith Yes, this is a general property of interpolation spaces $X_{\theta} = [X_0,X_1]_{\sigma,q}$ whenever $X_0 \hookrightarrow X_1$.
– ktoi
Commented Aug 27, 2023 at 16:55