  In a recent paper by the authors, growth properties of the Fourier transform
on Euclidean space and the Helgason Fourier transform on rank one symmetric
spaces of non-compact type were proved and expressed in terms of of a modulus
of continuity based on spherical means. The methodology employed first proved
the result on Euclidean space and then, via a comparison estimate for spherical
functions on rank one symmetric spaces to those on Euclidean space, we obtained
the results on symmetric spaces. In this note, an analytically simple, yet
overlooked refinement of our estimates for spherical Bessel functions is
presented which provides significant improvement in the growth property
estimates.
