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

作者：Fischer Veronique^*; Ricci Fulvio; Yakimova Oksana

来源：Journal of Functional Analysis, 2018, 274(4): 1076-1128.

DOI：10.1016/j.jfa.2017.09.014

• Fourier and spectral multipliers
• Analysis on nilpotent Lie groups
• Gelfand pairs and spherical transform
• Invariants

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(Sigma) of functions on the Gelfand spectrum Sigma of L-1(N)(K) which extend to Schwartz functions on R-d, once Sigma 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.

• 出版日期2018-2-15