On $L^2$-boundedness of pseudo-multipliers associated to the Grushin operator

TL;DR

This paper investigates the boundedness of pseudo-multiplier operators linked to the Grushin operator using spectral calculus, establishing Calderón–Vaillancourt-type theorems for these operators.

Contribution

It introduces a new class of pseudo-differential operators associated with the Grushin operator and proves boundedness results analogous to classical Calderón–Vaillancourt theorems.

Findings

Established $L^2$-boundedness for pseudo-multiplier operators

Extended Calderón–Vaillancourt theorem to Grushin setting

Provided spectral resolution-based operator definitions

Abstract

In this article we define analogues of pseudo-differential operators associated to the joint functional calculus of the Grushin operator using their spectral resolution, and study Calder\'on--Vaillancourt-type theorems for these operators.