$L\sp p$ bounds for spectral multipliers on nilpotent groups

Abstract

A criterion is given for the LP boundedness of a class of spec- tral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theo- rem of Hormander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction.