# Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs

# Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs

This chapter turns to a special class of Gelfland pairs, here called “nilpotent Gelfland pairs” for simplicity's sake and denoted by (*N*, *K*). It considers implications derived from, in principle, whenever we have a finite family of homogeneous, self-adjoint, commuting, left-invariant differential operators on a homogenous nilpotent Lie group *N*. A natural situation to consider is the one where the given operators are characterized by the property of being invariant under the action of a given compact group *K* of automorphisms of *N*. The general strategy of proof here is based on a bootstrapping argument, whose steps are determined by the level of complexity of the pairs involved, where the “complexity” depends on the structure of *N* and the way *K* acts on two layers of the Lie algebra.

*Keywords:*
Gelfland pairs, spectral multipliers, convolution kernels, nilpotent Gelfland pairs, bootstrapping argument, Heisenberg fan