For the p-circle generalized Bessel function J_0^p, the paper proves uniform O(|eta|^{-p/2}) decay on R^2 when 2/p is an integer greater than 2, and O(|eta|^{-1/2}) decay on compact sectors for all 0<p<1 and p=2.
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Asymptotic evaluations of generalized Bessel function of order zero related to the p-circle lattice point problem
For the p-circle generalized Bessel function J_0^p, the paper proves uniform O(|eta|^{-p/2}) decay on R^2 when 2/p is an integer greater than 2, and O(|eta|^{-1/2}) decay on compact sectors for all 0<p<1 and p=2.