In this talk we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem 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 operator M so that it satisfies weighted L^p estimates. As an application we prove L^p boundedness of Hermite pseudo-multipliers.