In standard spectral multiplier
theorems the critical index is determined by dimension of the
corresponding semigroup. If we consider elliptic operators this
dimension usually coincides with the Euclidean dimension of underlying
space. However for sub-elliptic operator the dimension of
the corresponding semigroup is strictly greater
than Euclidean dimension of underlying space.
Together with Michael Cowling we proved a Hörmander type spectral
multiplier theorem for a sub Laplacian on ${\rm SU}(2)$ with the critical
exponent equal to half the Euclidean dimension of the group.
This is an analogue of the result obtained by
Hebisch ,
Müller and
Stein
for the Heisenberg group. The group
${\rm SU}(2)$ is the only example of a simple Lie group
for which such precise spectral multiplier theorems
have been proved. This result is an important contribution
to our understanding of sub-elliptic operators.
We also developed techniques of proving very precise multiplier results
for operators with finite speed propagation of the corresponding wave equation.