Estimates for spectral projection operators of the sub-laplacian on the heisenberg group

Abstract

In this paper, we use Laguerre calculus to find theLP spectrum (λ, Μ) of the pair (L, iT). Here\({\cal L} = - \frac{1}{2}\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} \) md T = ∂/∂t with\(\left\{ {Z_1 ,...,Z_n ,\bar Z_1 ,,...,\bar Z_n , T} \right\}\)a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray λ > 0 in the Heisenberg brush and show that they are Calderon-Zygmund-Mikhlin operators. Estimates for these operators in L k p (Hn), HP(Hn), and S k pv (Hn) spaces can therefore be deduced.