# Bounding the propagation function

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 $$\alpha \in (0,1)-\{\frac{1}{2}\}$$.