Let $X$ be a Banach space (could work for over $\mathbb{R}$ as well?)


Let $A\colon D(A)\subset X\to X$ be a sectorial operator, and $e^{tA}$ be the semigroup generated by $A$.


It is well-known that for $m\in\mathbb{N}$ the real interpolation $(X,D(A^m))\_{\theta,p}$ ($\theta\in(0,1)$, $p\in[1,\infty]$) can be characterized as 
\begin{align}
(X,D(A^m))\_{\theta,p}:=
\left\{
x\in X
\
\,:\,
\int\_0^1 
\| t^{m(1-\theta)} A^m e^{tA}x \|\_X^p \frac{dt}t<\infty
\right\}.\tag{1}
\end{align}
See [Semi-Groups of Operators and Approximation](http://www.springer.com/us/book/9783642460685) by Butzer and Berens ($m=1$), or Triebel's book (Interpolation Theory, Function Spaces, Differential Operators)


Suppose further that $A$ is positive so that we can define fractional powers $A^\alpha$ ($\alpha\ge0$).


Question: do we have


\begin{align}
(X,D(A^\alpha))\_{\theta,p}=
\left\{
x\in X
\
\,:\,
\int\_0^1 
\|t^{\alpha(1-\theta)} A^\alpha e^{tA}x \|\_X^p \frac{dt}t<\infty
\right\}\ \ ?
\end{align}
Following the characterization/definition of the interpolation space, naively we would have 
$\int\_0^1 
\|t^{(1-\theta)} A^\alpha e^{tA^\alpha}x \|\_X^p \frac{dt}t<\infty$ as the condition. But we have a nice characterization (1) for (integer) powers of $A$. I wonder if a similar characterisation holds for the fractional powers.


The proof of (1) seems to use decompositions utilizing the integration by parts, e.g., for $m=2$, we split $x\in X$ as
$$
x=\int\_0^t sA^2 e^{sA} x ds+e^{tA}x-tA^{tA}x.
$$
In this sense, the derivative being of integer order seems to be essential in the proof of this direction. But can we somehow overcome this?



Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ generates?
Yes, there is a characterization like this. See Theorems 1-3 (p.182) in


*Markus Haase*, MR 2183483 [**A functional calculus description of real interpolation spaces for sectorial operators**](http://dx.doi.org/10.4064/sm171-2-4), *Studia Math.* **171** (2005), no. 2, 177--195.


The result you ask for is a special case treated in Section 7.2 (p. 191).


You find an online preprint version of the paper [here](http://fa.its.tudelft.nl/%7Ehaase/files/iposect-STUDIA.pdf). 


