In this article, we establish pointwise sparse domination results for Grushin pseudo-multipliers corresponding to various symbol classes, as a continuation of our investigation initiated in [BBGG21]. As a consequence, we deduce quantitative weighted estimates for these pseudo-multipliers.

In a recent work [BG21], first and third authors studied pseudo-multipliers associated to the Grushin operator, proving analogues of Calderón-Vaillancourt type theorems. In this article, we further extend the analysis of [BG21] and investigate quantitative weighted L p -boundedness for Grushin pseudomultipliers as well as for a family of operator-valued Fourier pseudo-multipliers in the context of the Grushin operator G. We establish pointwise domination of these operators by sparse operators. We also address analogous questions for pseudo-multipliers associated to a more general family of Grushin operators.