In Adams & Fournier's book "Sobolev Spaces" page 255, there are assertions:
$$ W^{s,p} = F^{s}_{p,2} \\ B^s_{p,p} = F^s_{p,p}$$
Note: Here $s>0$, $1\leq p<\infty$, and the fractional order Sobolev spaces are defined as complex interpolation spaces $W^{s,p} = [L^p , W^{m,p} ] _{s/m}$ where $m$ is the smallest integer greater than $s$.
AFAIK, the above definition is equivalent to the following definition (at least for $0<s<1$), employing Gagliardo seminorms: $f \in W^{s,p}$ iff $f \in W^{\left \lfloor{s}\right \rfloor ,p}$ and $[D^\alpha f] := (\int \frac{|D^{\alpha}f(x)-D^{\alpha}f(y)|^p } {|x-y|^{(s-\left \lfloor{s}\right \rfloor)p + n}} dxdy)^{1/p} < \infty$ for all $|\alpha|=\left \lfloor{s}\right \rfloor$.
The Triebel-Lizorkin spaces and the Besov spaces are defined by Paley-Littlewood decompositions.
In Triebel's book "Interpolation theory, function spaces, differential operators(1978)", page 169, the fractional order Sobolev spaces are defined as $W^{s,p} = B^s_{p,p}$ for $s>0$ non-integer.
Therefore, the two monographs are coherent only when $p=2$. Could someone explain about this situation?
My background: I have little experience and knowledge of function spaces. To get some knowledge, I am starting to have a brief overview of the famous monographs. However, the problem described above makes me very confused.
Any bits of help are welcome. Thank you!!
The notations in the two books are different.
With the definition using the Gagliardo seminorms, these are the real interpolation spaces (which are now more usually denoted by $W^{s,p}$ I think) when $s$ is not an integer $$ W^{s,p} = B^s_{p,p} = F^s_{p,p} = (L^p,W^{n,p})_{s/n,p} $$ while the complex interpolation gives you $$ H^{s,p} = F^s_{p,2} = [L^p,W^{n,p}]_{s/n} $$ with the seminorm $\left|f\right|_{H^{s,p}} = \|\Delta^{s/2}f\|_{L^p}$. The notation is quite misleading since when $n$ is an integer, then the classical $W^{n,p}$ is $H^{n,p}$ and not $B^n_{p,p}$.
By the general embeddings theorems between Besov and Triebel-Lizorkin spaces, you get $$ W^{s,p} ⊂ H^{s,p} $$ when $p≤ 2$ and the reverse inclusion when $p\geq 2$. Of course, when $p=2$, then $H^{s,2} = W^{s,2} = H^s$.