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

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 Calderón--Vaillancourt-type theorems for these operators.

Subsection 2.1 is new, discussing the notion of pseudo-multipliers for non-negative self-adjoint operators on $\sigma$-finite measure spaces. We show its relation with that for specific operators such as the Grushin operator. This discussion is particularly helpful in understanding our definition of symbol classes. Related changes occur in Subsection 1.3 too, resulting in an improved reading

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