Sharp estimates for imaginary powers of Bessel operators

Let $Lf(x):=-\frac{d^2}{d{x}^2}f(x)-\frac{r}{x}\frac{d}{dx}f(x),x>0,r>0$ be the Bessel operator on $((0,\mathrm{\infty }),\mathrm{\mid }\cdot \mathrm{\mid },x^{r}dx)$. In this paper, we prove the sharp weak type $(1,1)$ estimate for the imaginary power $L^{i\alpha },\alpha \in \mathbb{R}$, of the Bessel operator.