Hypoelliptic convolution operators and Rockland condition (Q1352326)

The authors study a left invariant, homogeneous, convolution operator \(K\) on a step two, simply connected homogeneous (nilpotent) Lie group. They prove that if \(K\) is hypoelliptic, then \(\pi(K)\) is injective on \(C^\infty(\pi)\) for every non-trivial irreducible unitary representation \(\pi\) of \(G\). The injectivity condition is commonly known as the ``Rockland condition'', and the converse of what the authors prove is commonly known as the Rockland conjecture.