  The paper deals with the problem under which conditions for the parameters
$s_1,s_2\in\mathbb{R}$, $1\leq p,q_1,q_2\leq\infty$ the Fourier transform
$\mathcal{F}$ is a nuclear mapping from $A^{s_1}_{p,q_1}(\mathbb{R}^n)$ into
$A^{s_2}_{p,q_2}(\mathbb{R}^n)$, where $A\in\{B,F\}$ stands for a space of
Besov or Triebel-Lizorkin type, and $n\in\mathbb{N}$. It extends the recent
paper arXiv:2112.04896 where the compactness of $\mathcal{F}$ acting in the
same type of spaces was studied.
