On radial functions and distributions and their Fourier transforms (Q485230)

The relationship between the Fourier transform in different dimensions of the same radial function, given in [\textit{L. Grafakos} and \textit{G. Teschl}, J. Fourier Anal. Appl. 19, No. 1, 167--179 (2013; Zbl 1311.42020)], is discussed. Spaces of radial test functions and radial tempered distributions \(S_{\mathrm{rad}}'(\mathbb R^n)\) are introduced and studied as is the operator \(J_{n,k}:S_{\mathrm{rad}}(\mathbb R^n)\to S_{ \mathrm{rad}}(\mathbb R^{k})\). The distributional operator \(\overline J _{n,k}\) is defined in Section~3 for radial distributions, and the case when \(n\) or \(k\) are 1 is discussed. Moreover, in Section~4 the Fourier transform related to the formula of two radial test functions by a certain operator \(Z_{n,k}\) is introduced. These are defined for distribution spaces for three cases in Section~5. Examples are given for the distribution extension of operators for radial distributions.