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

TL;DR

This paper extends the analysis of operator-valued Fourier pseudo-multipliers linked to Grushin operators, focusing on sparse domination and weighted estimates, building upon prior work in the field.

Contribution

It advances the understanding of Fourier pseudo-multipliers for Grushin operators by establishing new estimates and extending previous sparse domination results.

Findings

Established sparse domination for operator-valued Fourier pseudo-multipliers

Derived quantitative weighted estimates for these operators

Extended previous results to a broader class of Grushin-related operators

Abstract

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 $G = - \Delta_{x^{\prime}} - |x^{\prime}|^2 \Delta_{x^{\prime \prime}}$ on $\mathbb{R}^{n_1+n_2}.$