Oscillating spectral multipliers on groups of Heisenberg type

We establish endpoint estimates for a class of oscillating spectral multipliers on Lie groups of Heisenberg type. The analysis follows an earlier argument due to the second and fourth author but requires the detailed analysis of the wave equation on these groups due to M\"uller and Seeger. We highlight and develop the connection between sharp bounds for oscillating multipliers and the problem of determining the minimal amount of smoothness required for Mihlin-H\"ormander multipliers, a problem that was solved for groups of Heisenberg type but remains open for other groups.