Estimates on derivatives of Bessel function

Question

In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II" They have the following estimates for derivatives of Bessel functions: For $k \geq 2$ \begin{align} & z^{l}J^{(l)}_{k-1}(z) \ll z(1+z)^{-3/2} \\ & z^{l}Y^{(l)}_{0}(z) \ll (1+ |\log z|)(1+z)^{-1/2}\\ & z^{l}K^{(l)}_{0}(z) \ll (1+ |\log z|)(1+z)^{-1/2}. \end{align} For all z>0 and $l \geq 0$. Does anyone know about the proof of these? My major problem is the following: we have $$Y^{\prime}_0(z)=Y_1(z),$$ so using their second estimate we get$$Y_1(z)=Y^{\prime}_0(z) \ll \frac{(1+ |\log z|)}{z(1+z)^{1/2}}.$$ But we have the following asymptotic for Bessel functions $$Y_v(z) \sim \sqrt{\frac{2}{\pi z}}\sin(z-\frac{\pi}{4}- \frac{v\pi}{2}).$$

Answer 1

I think these estimates are wrong. They are true for $z\ll 1$, but for $z\gg 1$ the $z^l$ factors should be omitted. See, for example, Appendix C in Kowalski-Michel-VanderKam: Rankin-Selberg L-functions in the level aspect, Duke Math. J. 144 (2002), 123-191.

Added. I think this error does not cause any further problems. The estimates for nonzero $l$ are only used in (9.4), namely after differentiating (9.3) a few times in $z$ and $y$. Let us first assume that $z<Z$, and note that $y\ll Y=sN$ is automatic. The goal is to show that after each differentiation in $z$ (resp. $y$), we get similar expressions as the original (9.3) with extra factors of size $\ll P/z$ (resp. $P/y$). This seems fine due to the following formula for a Bessel function $B_\nu\in\{K_\nu,Y_\nu,J_\nu\}$: $$ 2t\frac{\partial}{\partial t}\{B_\nu(\alpha\sqrt{t})\}=\pm\alpha\sqrt{t}B_{\nu-1}(\alpha\sqrt{t})-\nu B_\nu(\alpha\sqrt{t}).$$ Note that here $\alpha\sqrt{t}$ plays the role of $4\pi\sqrt{xz}/c$ or $4\pi\sqrt{xy}/c$ in (9.3), i.e. $\alpha\sqrt{t}\ll P$. If $z>Z$ then this process is less effective for the $z$-differentiation, but then applying integration by parts in $x$ several times in the original (9.3) allows to gain many extra negative powers in $z$. I have not worked out this last step, but it seems ok. Contacting the authors would be a good idea as well.

Answer 2

you can see http://arxiv.org/abs/1408.5652v1 for some reference