I was studying the paper "Two-weight mixed norm estimates for a generalized spherical mean Radon transform acting on radial functions" by Ciaurri et al.
The authors of this paper want to get $L^p-L^q$ boundedness of the spherical mean value operator (for radial functions). To this end, they study a family of operators defined for $\alpha > - \frac{1}{2}$, and $\beta > - \alpha - \frac{1}{2}$, given by
$$M^{\alpha, \beta}f (x, t) = \int\limits_{0}^{\infty} K^{\alpha, \beta}_t (x, z) f(z) z^{2\alpha - 1} \mathrm{d}z,$$
where
$$K^{\alpha \beta}_t(x, z) = 2^{\alpha + \beta} \Gamma (\alpha + \beta + 1) \int\limits_{0}^{\infty} \frac{J_{\alpha + \beta} (ty) J_{\alpha} (xy) J_{\alpha} (zy)}{(ty)^{\alpha + \beta} (xy)^{\alpha} (zy)^{\alpha}} y^{2\alpha - 1} \mathrm{d}y.$$
Here, $J_{\nu}$ is the Bessel function of first kind. To get an estimate of the operator $M^{\alpha, \beta}$, the authors try to get estimates of the kernel $K^{\alpha, \beta}_t$. For that, they first derive a formula for the integral involved in the kernel. We have
$$\int\limits_{0}^{\infty} \frac{J_{\alpha + \beta} (ty) J_{\alpha} (xy) J_{\alpha} (zy)}{(ty)^{\alpha + \beta} (xy)^{\alpha} (zy)^{\alpha}} y^{2\alpha - 1} \mathrm{d}y = \frac{(xz)^{\alpha + \beta - 1}}{\sqrt{2\pi} t^{\alpha + \beta}} \begin{cases} 0, & \text{if } t < x + z. \\ (\sin v)^{\alpha + \beta - \frac{1}{2}} \mathbf{P}^{\frac{1}{2} - \alpha - \beta}_{\alpha - \frac{1}{2}} ( \cos v), & \text{if } | x - z | < t < x + z. \\ \frac{2}{\Gamma(\beta)} (\sinh u)^{\alpha _ \beta - \frac{1}{2}} \mathbf{Q}^{\frac{1}{2} - \alpha - \beta}_{\alpha - \beta} (\cosh u), & \text{if } t > x + z. \end{cases}$$
Here, $\mathbf{P}^{\mu}_{\nu}$ is the Ferrers function of the first kind, and $\mathbf{Q}^{\mu}_{\nu}$ is the Olver's function (which is defined in terms of the associated Legendre's function of the second kind). One can find these functions in the book: "NIST Handbook of Mathematical Functions", by Olver et al.
Coming to Ciaurri's ideas, we first want to get the asymptotic behaviour of $\mathbf{P}^{\mu}_{\nu}$ at $-1^+$ and $1^-$, which are the end-points of the domain of definition of $\mathbf{P}^{\mu}_{\nu}$. Similarly, we will get the asymptotic behaviour of $\mathbf{Q}^{\mu}_{\nu}$ at the end-points $1^+$ and $\infty$. While most calculations done by Ciaurri et al. and me match, I am not sure how to address one issue (each of the two functions). Before I give my questions, here are the asymptotic behaviours computed by me (using the formulae from Olver's book, and properties of the hypergeometric functions).
Near $-1^+$, we have
$$\mathbf{P}^{\frac{1}{2} - \alpha - \beta}_{\alpha - \frac{1}{2}} (x) = \begin{cases} \frac{2^{\frac{1}{2} \left( \alpha + \beta - \frac{1}{2} \right)} \Gamma \left( \alpha + \beta - \frac{1}{2} \right)}{\Gamma \left( \beta\right) \Gamma (2 \alpha + \beta)} \left( 1 + x \right)^{\frac{1}{2} \left( \frac{1}{2} - \alpha - \beta \right) }, & \text{if } \alpha + \beta > \frac{1}{2}, - \beta \notin \mathbb{N}, 1 - 2 \alpha - \beta \notin \mathbb{N}. \\ \frac{1}{\pi} \sin \left( \pi \left( \alpha - \frac{1}{2} \right) \right) \ln \left( \frac{1 + x}{2} \right), & \text{if } \alpha + \beta = \frac{1}{2}, - \beta \notin \mathbb{Z}. \\ \frac{1}{\pi} 2^{\frac{1}{2} \left( \frac{1}{2} - \alpha - \beta \right) } \Gamma \left( \frac{1}{2} - \alpha - \beta \right) \sin \left( \pi \left( \alpha - \frac{1}{2} \right) \right) \left( 1 + x \right)^{\frac{1}{2} \left( \alpha + \beta - \frac{1}{2} \right) }, & \text{if } \alpha + \beta < \frac{1}{2}, \alpha - \frac{1}{2} \notin \mathbb{Z}. \end{cases}$$
Similarly, at near $1^+$, I get
$$\mathbf{Q}^{\frac{1}{2} - \alpha - \beta}_{\alpha - \frac{1}{2}} (x) = \begin{cases} \frac{2^{\frac{1}{2} \left( \alpha + \beta - \frac{1}{2} \right) - 1} \Gamma \left( \alpha + \beta - \frac{1}{2} \right)}{\Gamma (2 \alpha + \beta )} \left( x - 1 \right)^{\frac{1}{2} \left( \frac{1}{2} - \alpha - \beta \right)}, & \text{if } \alpha + \beta > \frac{1}{2}. \\ - \frac{1}{\Gamma \left( \alpha + \frac{1}{2} \right)} \ln \left( 2 ( x - 1 ) \right), & \text{if } \alpha + \beta = \frac{1}{2}. \\ \frac{2^{\frac{1}{2} \left( \frac{1}{2} - \alpha - \beta \right) + 1} \Gamma \left( \frac{1}{2} - \alpha - \beta \right)}{\Gamma \left( 1 - \beta \right)} \left( x - 1 \right)^{\frac{1}{2} \left( \alpha + \beta - \frac{1}{2} \right)}, & \text{if } \alpha + \beta < \frac{1}{2}. \end{cases}$$
Based upon these calculations, I have the following questions:
For the so-called "exceptional cases" (those excluded in the calculations above) for the Ferrers function ($\mathbf{P}$), how can we get the asymptotic behaviour? As such it seems that in these cases we get zero. However, asymptotically, how can one guess which function is similar to $\mathbf{P}$?
I do not get any "exceptional cases" for the Olver's function $\mathbf{Q}$, while Ciaurri et al. do get them! Are the asymptotic behaviours found by me wrong?