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For which $t$ and $p$ does function of the form $$ f(x) = 1/|x|^{\alpha} ,\quad 0<\alpha<1$$ belong to the fractional Sobolev space $H_p^t(\mathbb{R})$, $p>1,t\geq 0$?

UPD: actually I am intersted in $$ f(x) = \chi_{[-1,1]}/|x|^{\alpha} ,\quad 0<\alpha<1$$ where $\chi_{[-1,1]}$ is a characteristic function of the interval $[-1,1]$.

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  • $\begingroup$ What are your thoughts on the problem? $\endgroup$
    – Silvia Ghinassi
    Apr 27, 2016 at 14:56
  • $\begingroup$ @SilviaGhinassi I have changed a bit the question, maybe not it looks more reasonable. $\endgroup$
    – demitau
    Apr 27, 2016 at 15:07
  • $\begingroup$ @SilviaGhinassi Basically I just don't know how to compute the inverse Fourier transform of $(1+|\xi|^2)^{t/2} |\xi|^{-1+\alpha}$ $\endgroup$
    – demitau
    Apr 27, 2016 at 15:30

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