On some operator-valued Fourier pseudo-multipliers associated to Grushin operators

This is a continuation of our work [BBGG23, BBGG22] where we have initiated the study of sparse domination and quantitative weighted estimates for Grushin pseudo-multipliers. In this article, we further extend this analysis to study analogous estimates for a family of operator-valued Fourier pseudo-multipliers associated to Grushin operators $G=-{\mathrm{\Delta }}_{x^{\mathrm{\prime }}}-\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2{\mathrm{\Delta }}_{x^{\mathrm{\prime }\mathrm{\prime }}}$ on ${\mathbb{R}}^{n_1+n_2}\mathrm{.}$