Weighted norm inequalities for Weyl multipliers and fourier multipliers on the Heisenberg group

In this paper, we prove 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 R-boundedness of the derivative of the multiplier is not necessary for boundedness of the multiplier transform.