### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

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

**Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs.** / Fischer, Véronique; Ricci, Fulvio; Yakimova, Oksana.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 274, no. 4, pp. 1076-1128. https://doi.org/10.1016/j.jfa.2017.09.014

}

TY - JOUR

T1 - Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

AU - Fischer, Véronique

AU - Ricci, Fulvio

AU - Yakimova, Oksana

PY - 2018/2/15

Y1 - 2018/2/15

N2 - 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.

AB - 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.

KW - Analysis on nilpotent Lie groups

KW - Fourier and spectral multipliers

KW - Gelfand pairs and spherical transform

KW - Invariants

KW - Primary

KW - Secondary

UR - http://www.scopus.com/inward/record.url?scp=85031118910&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2017.09.014

DO - 10.1016/j.jfa.2017.09.014

M3 - Article

VL - 274

SP - 1076

EP - 1128

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 4

ER -