Asymptotic evaluations of generalized Bessel function of order zero related to the p-circle lattice point problem

Let $p$ and $r$ be positive real numbers. Then, we consider the lattice point problem of the closed curve $p$-circle $\{x\in {\mathbb{R}}^2\mathrm{\mid }\mathrm{\mid }{x}_1{\mathrm{\mid }}^p+\mathrm{\mid }{x}_2{\mathrm{\mid }}^p=r^p\}$ which is a generalization of the circle ($p=2$). Following the harmonic analytic approach of S. Kuratsubo and E. Nakai for the case of a circle, we need to investigate properties of appropriately generalized Bessel functions for $p$ in order to tackle the problem. Thus, in this paper, we derive asymptotic evaluations of the generalized Bessel function of order zero, such as uniformly asymptotic estimates on compact sets on quadrants of ${\mathbb{R}}^2$ for the cases $0<p\le 1$ or $p=2$, and, as stronger results, uniformly asymptotic estimates on ${\mathbb{R}}^2$ for the cases $p$ such that $\frac{2}{p}$ are the natural numbers.