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A function $f: S^1\to S^1$ is of class $H^{1/2}(S^1,S^1)$, when $$\int\limits_{S^1} \int \limits_{S^1} \frac{|f(x)-f(y)|^2}{|x-y|^2} dS(x)dS(y)< \infty$$

but why is that equivalent to $$\sum\limits_{j=-\infty}^{\infty}|j| |a_j|^2<\infty$$ where $a_j$ are the Fourier coefficients of $f$?

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    Oh, sorry yes, just a typo. I corrected it – 2017-01-22
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    Have you tried using Plancherel's theorem on the first expression? – 2017-01-23
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    It is enough to show that $$\iint_{S^1\times S^1}\left|\frac{e^{nix}-e^{niy}}{x-y}\right|^2\,d\mu$$ behaves like $|n|$ for large $n$s. For such a purpose, one may employ Laplace's method or other techniques, like separating the cases in which $x$ and $y$ are close from the cases in which $x$ and $y$ are fairly distant. – 2017-02-07

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