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 [Springer INdAM Ser., vol. 45 (2021)], but requires the detailed analysis of the wave equation on these groups due to Müller and Seeger [Anal. PDE 8 (2015)]. We highlight and develop the connection between sharp bounds for oscillating spectral multipliers and the problem of determining the minimal amount of smoothness required for Mihlin–Hörmander multipliers, a problem that has been solved for groups of Heisenberg type but remains open for other groups.