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 Muller 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-Hormander multipliers, a problem that was solved for groups of Heisenberg type but remains open for other groups.

[1]  R. Melrose.  Propagation for the Wave Group of a Positive Subelliptic Second-Order Differential Operator , 1986 .

[2]  Alessio Martini,et al.  Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type , 2018, Journal of the European Mathematical Society.

[3]  A. Hulanicki.  A functional calculus for Rockland operators on nilpotent Lie groups , 1984 .

[4]  E. Stein,et al.  On spectral multipliers for Heisenberg and related groups , 1994 .

[5]  M. Christ.  bounds for spectral multipliers on nilpotent groups , 1991 .

[6]  Alessio Martini.  Analysis of joint spectral multipliers on Lie groups of polynomial growth , 2010, 1010.1186.

[7]  E. Stein,et al.  $L^p$-estimates for the wave equation on the Heisenberg group , 1999 .

[8]  Alessio Martini,et al.  $L^p$ spectral multipliers on the free group $N_{3,2}$ , 2012, 1210.8090.

[9]  W. Hebisch.  Multiplier theorem on generalized Heisenberg groups , 1993 .

[10]  Alessio Martini.  Spectral multipliers on Heisenberg–Reiter and related groups , 2012, 1212.0775.

[11]  Alessio Martini,et al.  Spectral multipliers on 2-step groups: topological versus homogeneous dimension , 2015, 1508.01687.

[12]  Geometric Aspects of Harmonic Analysis , 2021, Springer INdAM Series.

[13]  M. Christ.  Ü BOUNDS FOR SPECTRAL MULTIPLIERS ON NILPOTENT GROUPS , 2009 .

[14]  Paolo Ciatti,et al.  Strongly Singular Integrals on Stratified Groups , 2018, Geometric Aspects of Harmonic Analysis.

[15]  A. Seeger,et al.  Sharp Lp bounds for the wave equation on groups of Heisenberg type , 2014, 1408.3051.