### Abstract

It has been shown [1,2,9,10] that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L
^{1}(N)
^{K} commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)
^{K} of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L
^{1}(N)
^{K} which extend to Schwartz functions on R
^{d}, once Σ is suitably embedded in R
^{d}. We call this property (S). We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N. We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.

Original language | English |
---|---|

Pages (from-to) | 1076-1128 |

Number of pages | 53 |

Journal | Journal of Functional Analysis |

Volume | 274 |

Issue number | 4 |

Early online date | 28 Sep 2017 |

DOIs | |

Publication status | Published - 15 Feb 2018 |

### Keywords

- Analysis on nilpotent Lie groups
- Fourier and spectral multipliers
- Gelfand pairs and spherical transform
- Invariants
- Primary
- Secondary

### ASJC Scopus subject areas

- Analysis

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## Profiles

### Veronique Fischer

- Department of Mathematical Sciences - Senior Lecturer
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)

Person: Research & Teaching

## Cite this

*Journal of Functional Analysis*,

*274*(4), 1076-1128. https://doi.org/10.1016/j.jfa.2017.09.014