\(L^ p\) bounds for spectral multipliers on nilpotent groups. (English) Zbl 0739.42010

Christ, Michael

Trans. Am. Math. Soc. 328, No. 1, 73-81 (1991).

For a class of spectral multiplier operators associated to left-invariant homogeneous subelliptic second-order differential operators on nilpotent Lie groups, a criterion of Hörmander type for the \(L^ p\)-and weak type (1,1)-boundedness is given. The order of differentiability required is half the homogeneous dimension of the group.

Reviewer: J.Marschall (Neubiberg)

MSC:

42B15	Multipliers for harmonic analysis in several variables
43A22	Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
35P99	Spectral theory and eigenvalue problems for partial differential equations

Keywords:

Fourier multiplier operator; heat kernel; spectral multiplier operators; subelliptic second-order differential operators; nilpotent Lie groups; criterion of Hörmander type