A restriction theorem for the H-type groups

Abstract

We prove that the restriction operator for the H-type groups is bounded from L p L^p to L p ′ L^{p’} for p p near to 1 1 when the dimension of the center is larger than one, and the range of p p depends on the dimension of the center. This is different from the Heisenberg group, on which the restriction operator is not bounded from L p L^p to L p ′ L^{p’} unless p = 1 p=1 .

A functional calculus and restriction theorem on H-type groups

“…Let L = +∞ 0 λdE(λ) be the spectral decomposition of L. Then the restriction operator can be formally written P λ f = δ λ (L)f = lim ǫ→+∞ χ (λ−ǫ,λ+ǫ) (L)f which is well defined for a Schwartz function f , where χ (λ−ǫ,λ+ǫ) is the characteristic function of the interval (λ − ǫ, λ + ǫ). Liu and Wang [5] investigated the restriction theorem for the sublaplacian L on H-type groups with the center whose dimension is more than 1. They give the following result:…”

H-type group, a-Weyl transform and pseudo-differential operators

“…where (x, y, t) ∈ H, a ∈ R m * = R m \{0} and φ ∈ L 2 (R n ). Thus π a is the irreducible representation on H. For more details we refer the reader to [10] and [16]. π a (x, y, t) can also be written as π a (x, y, t) = e ia•t π a (x, y), where…”

Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group

“…Other results extending the restriction theorem of Müller to more general nilpotent groups through spectral analysis have been considered in [20,21] and [35,36]. Finally, let us mention that applications of non commutative Fourier analysis have been also used to study the heat equation associated to sublaplacians on groups, see for instance [1].…”