When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and now ask how singular are they, i.e. what is the largest exponent $s$ such that this function/distribution belongs to $H^s$. For simplicity let me restrict to the interval $[-\pi,\pi]$ where the critical $s$ such that $f \in H^{-s}$ (note the minus sign) can be found by investigating the convergence of the series $\sum_{n} \frac{|c_n|^2}{n^{2s}}$ where $c_n$ are Fourier coefficients.
Example 1 Let us take $f=\chi_{[0,1]}$. Then after some computations this boils down to the convergence of $\sum_{n} \frac{\sin^2{n}}{n^{2+2s}}$ and thus we get that $f \in H^s$ for $s<\frac12$ but $f \notin H^\frac{1}{2}$.
Example 2 If we take the Dirac delta distribution $\delta$ then the sequence of Fourier coefficients is constant and one gets that $\delta \in H^s$ for $s<-\frac12$ but $\delta \notin H^{\frac12}$.
Example 3 Let us now take $f(x)=\frac{1}{\sqrt{|x|}}$. Then $f \notin L^2=H^0$ but $f \in H^s$ for $s<0$
So for these examples the critical exponents are $s_0=\frac12,-\frac12,0$ respectively. In all these cases $f \notin H^{s_0}$ but $f \in H^s$ for $s<s_0$.
Question Is it possible that for a critical exponent as above in fact $f \in H^{s_0}$ (but $f \notin H^s$ for $s>s_0$?)?
This should follow from general facts concerning Dirichlet series but I'm not an expert in this field and I will be happy if someone could give me an answer
