Is $B^s_{p,p} = W^{s,p}$?

Question

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!!

Answer 1

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 $H^{s,2} = W^{s,2} = H^s$.