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For $\alpha \in (0,1)$, set $\omega:\mathbb{R^+}\times \mathbb{R^+} \to \mathbb{R}$ defined as following

$$\omega(t;\tau):=1-\pi^{-1}\int_0^\infty \frac{e^{-rt-\tau r^\alpha \cos(\alpha \pi)}\sin(\tau r^{\alpha}\sin(\alpha \pi))}{\pi r}dr, \ t, \tau>0$$ Can we prove that exist $a$, $b$, $c$, $\alpha \in (0,1)$, $K>0$, such that $b>-1$, $a+b+c=-1$ and

$$|v(t;\tau)|=\left|-\frac{\partial}{\partial \tau}\omega(t;\cdot)\right|\leq K(t-\tau)^a\tau^bt^c, \ \forall t,\tau >0, t-\tau>0?$$ I got observe that for $\alpha=1/2$ its false ploting on software Maple, but the Maple couldn't calculate the integral equation for $\alpha \in (0,1)-\{\frac{1}{2}\}$.

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