Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$. Here, $F$ denotes the Fourier transform. Such spaces are known to admit Littlewood-Paley characterisation, i.e. one can esitmate the norm $\|\phi\|_{H^s_p}$ by $$ \| ( \sum_{n\geq 0} 4^{sn} |\phi_n|^2 )^{1/2} \|_{L_p},$$ where $\phi_n(x) = F^{-1}(\psi_n(\xi)F(\phi))(x)$ for $\psi_n(x)$ being a smooth function that is equal to $1$ on the ring $2^{n-1}<|\xi|<2^{n}$ and $0$ outside a neighborhood of it.
Above is the definition of the Sobolev space for $\mathbb R$. It is well known that using local charts one can define Sobolev spaces on manifolds, and get the Littlewood-Paley characterisation as well.
It is also well known that one can define Fourier transform directly on the circle and get a Fourier series as a result, that are somewhat easier to work with.
Now the question: is it known that if I define fractional Sobolev spaces on the circle directly (without charts), it will have a Littlewood-Paley characterisation of such space? If yes, what would be a reference?
