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Suppose that $\theta(t,x)$ is even about $x$ and is smooth. $0\le \gamma<1/2$, $0<\delta<1-2\gamma$. $\Lambda=(-\partial_{xx})^{1/2}$

My Question: How to prove that $$ \int_0^{\infty} \frac{(\Lambda^{\gamma}\theta)(t,x)-(\Lambda^{\gamma}\theta)(t,0)}{x^{1+\delta}}dx=\int_0^{\infty}\frac{\theta(t,x)-\theta(t,0)}{x^{1+\delta+\gamma}}dx $$ My efforts: Since $\theta$ is even, $\Lambda^{\gamma}\theta$ is even. Using parseval's theorem, we can obtain $$ \int_0^{\infty} \frac{(\Lambda^{\gamma}\theta)(t,x)-(\Lambda^{\gamma}\theta)(t,0)}{x^{1+\delta}}dx=\frac{1}{2}\int_{-\infty}^{\infty}\frac{(\Lambda^{\gamma}\theta)(t,x)-(\Lambda^{\gamma}\theta)(t,0)}{|x|^{1+\delta}}dx=\frac{1}{2}\int_{-\infty}^{\infty}( \theta(t,x)-\theta(t,0))\Lambda^{\gamma}(|x|^{-(1+\delta)})dx $$ However, $\Lambda^{\gamma}(|x|^{-(1+\delta)})$ have nonsense.

This question is from the last proof of this paper. Thanks for any help! Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation

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    $\begingroup$ It looks like this is a simple reformulation of the identity $(-\partial_{xx})^{\delta/2} (-\partial_{xx})^{\gamma/2} f(0) = (-\partial_{xx})^{(\delta+\gamma)/2} f(0)$, where $f(x) = \theta(t,x)$, but an appropriate constant is missing in the right-hand side. (Note: nothing seems to depend on $t$ here.) $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 11, 2021 at 21:14
  • $\begingroup$ Thanks for your comment. I get an answer in MSE. math.stackexchange.com/questions/4195771/… $\endgroup$
    – Mr.xue
    Commented Jul 12, 2021 at 1:36

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