I study Plancherel measure for
Laplace operators on a nilpotent Lie group.
Such a measure
$\mu$, introduced by Christ, satisfies the following formula
$\|K_{F(L)}\|^2_{L^2}= \int_0^{\infty}|F(\lambda)|^2\;d\mu (\lambda),$
where $K_{F(L)}$ is the kernel of the multiplier operator $F(L)$
of a self-adjoint operator $L$ defined on a Lie group.
The Plancherel measure gives us a precise description of the
$L^2 \to L^{\infty}$ norms of the spectral projectors of the operator $L$.
These norms of spectral projectors were investigated in the case of an elliptic
operator on a compact manifold by Hörmander and Sogge.
I proved the smoothness of the Plancherel measure for a certain class of
Laplace operators and showed that this leads to an improvement
of Alexopoulos' multiplier theorem.