# Interpolation for Sobolev spaces

How one can identify the following (complex) interpolation space $$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$ where $$\Omega$$ is a regular domaine. After research, it seems that this depend on the position of $$\theta \in (0,1)$$:

for $$0<\theta<1/4$$, $$E_\theta=H^{2\theta}(\Omega)$$ and for $$1/4 <\theta <1$$ we have $$E_\theta=H^{2\theta}_0(\Omega)$$, while the case $$\theta =1/4$$ is critical.

Some inclusions are immediate while the others are not. Is there any elegant way to establish such identification?

Any reference would be helpful.

• There is a answer to a related question with some info, maybe this is sufficiently elegant.. Jul 29, 2020 at 7:02
• @Hannes thank you, but this gives just the link to domains of fractional powers and not the explicit characterization depending on $\theta$. Aug 17, 2020 at 19:28

## 1 Answer

This is a result by R. Seeley: Interpolation in Lp with boundary conditions, Studia Mathematica, 1972.

The main ingredient of the proof is that step functions are pointwise multipliers in Hs for s<1/2.

• Thank you, but it seems that the result was already proved by many others, e.g., Fujiwara, Grisvard... I'm looking for an elementary way to do it. Mar 3, 2020 at 6:33
• The book of Lions and Magenes is another reference for this kind of result. But what do you mean by "elementary"? Surely not using pre-calculus only? Mar 26, 2021 at 2:48
• @MichaelRenardy I meant not using heavy theories such as $H^\infty$ calculus, or a way that can be generalized to more complex spaces e.g., product spaces. Aug 5, 2022 at 1:54