We establish Boas-type equivalence theorems for Fourier–Laplace series on the unit sphere in the mixed-norm spaces \(\mathcal {S}^{(p,q)}(\sigma ^{m-1})\) , which provide natural spherical analogues of the Wiener algebra and Stepanets classes. We characterize functions in the generalized smoothness classes \(H_{p,q}^{r,\varphi }(\sigma ^{m-1})\) and \(h^{r,\varphi }_{p,q}(\sigma ^{m-1})\) using weighted summability conditions on their Fourier–Laplace coefficients. These results extend the classical Boas theorem to the spherical mixed-norm setting, with the modulus of continuity measured in the \(\mathcal {S}^{(p,q)}\) norm.
