On Hermite pseudo-multipliers

In this article we deal with a variation of a theorem of Mauceri concerning the $L^p$ boundedness of operators $M$ which are known to be bounded on $L^2.$ We obtain sufficient conditions on the kernel of the operaor $M$ so that it satisfies weighted $L^p$ estimates. As an application we prove $L^p$ boundedness of Hermite pseudo-multipliers.