Let $(H, \|\cdot\|)$ be a Hilbert space, $A \colon D(A)\subset H \longrightarrow H$ generates an analytic semigroup $T(t)$ on $H$. We define the following Banach space with the respect norm
$$F=\{x\in H : \int\_0^{+\infty} \|AT(t)x\|^2 dt <\infty\},$$
$$\|x\|\_F =\|x\|+ \left(\int\_0^{+\infty} \|AT(t)x\|^2 dt\right)^{1/2}.$$


If we consider $H=L^2(\Omega)$ and $D(A)=H^2(\Omega)\cap H\_0^1(\Omega)$ ($\Delta$ for example). Is there any relation between the space $F$ and Besov spaces, or can we prouve the interpolation $F=(H,D(A))\_{1/2,2}=H\_0^1(\Omega)$ ?


Thank you for any help.



Relation between a norm and norm of Besov spaces
Yes, your identities are correct. Theorem 1.14.5 in Triebel's book [T] says that $$F = (H,D(A))\_{1/2,2},$$ and $(1/2,2)$-real interpolation spaces between Hilbert spaces are in fact exactly the $1/2$-complex interpolation spaces, so $$(H,D(A))\_{1/2,2} = [H,D(A)]\_{1/2}$$ as proven for example in [P]. If the operator $A$ now admits bounded imaginary powers, then you have $$[H,D(A)]\_{1/2} = D(A^{1/2}),$$
see e.g. [T, Theorem 1.15.3]. The latter is indeed known for the Laplacian and there you will also indeed have $D(A^{1/2}) = H^1\_0(\Omega)$. See also [this related question](https://mathoverflow.net/questions/280587/interpolation-spaces).




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[T] *Triebel, Hans*, Interpolation theory, function spaces, differential operators. Leipzig: Barth. 532 p. (1995). [ZBL0830.46028](https://zbmath.org/?q=an:0830.46028).


[P] *Peetre, Jaak*, Sur la transformation de Fourier des fonctions à valeurs vectorielles. Rendiconti del Seminario Matematico della Università di Padova, Volume 42 (1969), p. 15-26


