Stein's square function associated with the Bochner-Riesz means on M\'etivier groups and its applications

In this paper, we study the $L^p$-boundedness of Stein's square function $\mathfrak{S}^{\alpha}(\mathcal{L})$ associated with the sub-Laplacian $\mathcal{L}$ on M\'etivier group $G$. A key aspect of our result is that the smoothness condition is expressed in terms of the topological dimension $d$ of the underlying M\'etivier group $G$. Consequently, we also present several applications of the $L^p$-boundedness of $\mathfrak{S}^{\alpha}(\mathcal{L})$. First, we provide an alternate proof of the sharp $L^p$-boundedness result for spectral multipliers on M\'etivier groups, recently obtained by Niedorf [Niedorf, Studia Math., 2025]. Next we prove $L^p$-boundedness of maximal spectral multipliers and consequently establish sharp $L^p$-boundedness result for the maximal Bochner-Riesz operator on M\'etivier groups, which also yields pointwise almost everywhere convergence of Bochner-Riesz means with smoothness parameter given in terms of the topological dimension of $G$. In case of M\'etivier groups our result improves upon the existing works of Mauceri-Meda [Mauceri, Meda, Rev. Mat. Iberoam., 1990] and Horwich-Martini [Horwich, Martini, J. Lond. Math. Soc., 2021]. Our result further imply the mixed norm regularity estimates for the solution of fractional Schr\"odinger equation on M\'etivier groups, where the regularity index is again expressed in terms of the topological dimension of $G$. Finally, we study the $L^{p_1}(G) \times L^{p_2}(G)$ to $L^p(G)$ boundedness of the bilinear Bochner-Riesz means and its maximal version, associated with the sub-Laplacian on M\'etivier group $G$. Our result improves upon the recent work of the author with Bagchi and Molla [Bagchi, Molla, Singh, J. Funct. Anal., 2026] in the range $2\leq p_1, p_2 <\infty$. In the same range, ......