The author studies analytic and Gevrey hypoellipticity of operators of the type \[ P= \sum C_{\alpha\beta\gamma} y^\alpha D^\beta_x D^\gamma_y, \] where \(x\in \mathbb{R}^n\), \(y\in\mathbb{R}^m\), and the sum is finite, satisfying suitable conditions as in \textit{V. V. Grušin} [Math. USSR Sbornik 17, 497-514 (1972; 255.35022)]. As well-known, the hypoellipticity of \(P\) depends on the injectivity on the Schwartz space \({\mathcal S}\) of auxiliary globally elliptic operators, of the form \[ L= \sum a_{\alpha\beta} y^\alpha D^\gamma_y. \] Basic step of the paper is a more precise analysis of \(L\) in the frame of the spaces \({\mathcal S}^r_s\) of Gel'fand and Shilov. From this, precise results of Gevrey hypoellipticity are deduced for \(P\). As examples, the author treats the second-order operators \[ D^2_y+ y^{2k} D^2_{x_1}+ D^2_{x_2},\quad D^2_y+ y^{2k} D^2_{x_1}+ y^{2h} D^2_{x_2}\quad\text{in }\mathbb{R}_y\times \mathbb{R}^2_{x_1,x_2}. \] We address to the survey paper of \textit{D. Tartakoff} [in Microlocal analysis and spectral theory, Kluwer, Dordrecht etc. NATO ASI Ser. C, 490, 39-59 (1997; Zbl 0879.35167)], for comparison with recent results on sum-of-squares operators.