It is shown that the Riesz–Bessel transformations \(R_{B_j}\), \(j=1,2,\dots, n-1, \) are bounded on the space \(B_a\), where \[ B_a= \{ f: f \in \cap_{m=1}^{\infty} B_m, R_f>0 \}, \]
\[ \| f\| _{B_a}=\inf_{\alpha>0}\{ 1/\alpha: I(f,\alpha)\leq 1\}, \]
\[ B_m= L_{p_m,\nu} (\mathbb{R}_n^+)=\left\{ f: \| f\| _{ L_{p_m, \nu}}\equiv \bigg(\int_{\mathbb{R}^+_n} | f(x)| ^{p_m} x_n^{2 \nu} dx \bigg)^{1/p_m} < \infty \right\} \] and \(R_f\) is the radius of the convergence of the series \[ I(f,\alpha)=\sum_{m=1}^{\infty}a_m \| f\| _{B_m}^m \alpha^m, \] if and only if there exist constants \(\alpha\) and \(\beta\) such that the sequence \(\{p_m\}\) satisfies \(1<\alpha<p_m<\beta\) for all \(a_m\neq 0\).