arXiv subjects
Publications and source records attributed to Veronique Fischer.
In this paper, we present first results of our investigation regarding symbolic pseudo-differential calculi on nilpotent Lie groups. On any graded Lie group, we define classes of symbols using difference operators. The operators are obtained from these symbols via the natural quantisation given by the representation theory. They form an algebra of operators which shares many properties with the usual Hormander calculus.
In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.
Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a compact group of automorphisms of N, and the algebra D(N)^K of left-invariant and K-invariant differential operators on N is commutative. In these hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies Vinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N). Fixing a system D of d formally self-adjoint generators of D(N)^K, the Gelfand spectrum of the commutative convolution algebra L^1(N)^K can be canonically identified with a closed subset S_D of R^d. We prove that, on a nilpotent Gelfand pair satisfying Vinberg's condition, the spherical transform establishes an isomorphism from the space of $K$-invariant Schwartz functions on N and the space of restrictions to S_D of Schwartz functions in R^d.
In this short note, we determine the spectrum of the Heisenberg oscillator which is the operator defined as $L + |z|^2$ on the 3-dimensional Heisenberg group $H_1=\mathbb C \times \mathbb R$ where $L$ stands for the positive sublaplacian and $z$ for the complex variable of a point of the group.
For p odd, the Lie group SO_0(p+1,p+1) has a family of unitary degenerate principal series representations realized on the space of real (p+1) by (p+1) skew symmetric matrices, similar to the Stein's complementary series for SL(2n,C) or Speh's representation for SL(2n,R). We consider their restriction on the subgroup G= SO(p+1,p) and prove that they are still irreducible and is equivalent to (a unitarization of) the principal series representation of G, and also irreducible under a maximal parabolic subgroup of G.
This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. We prove a result, of independent interest for the representation theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as a direct sum of [n,n] and a K-invariant irreducible subspace.
In this paper, we give the expressions for the bounded spherical functions, or equivalently the spherical functions of positive type, for the free two-step nilpotent Lie groups endowed with the actions of orthogonal groups or their special subgroups. Next we deduce some results about the (Kohn) sub-Laplacian, and we compute the radial Plancherel measure.
The spectrum of a Gelfand pair of the form (K lx N, K), where N is a nilpotent group, can be embedded in a Euclidean space Rd . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on Rd has been proved already when N is a Heisenberg group and in the case where N = N3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11]. We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K lx N, K) in Vinberg's list [27, 30]. (The references numbers refers to the bibliography at the end of the article)
The aim of my PhD work is to study the $L^p$-boundedness of operators on two classes of two-step nilpotent Lie groups, using Plancherel formulas and spherical functions as tools. The first class of groups consists of the groups of Heisenberg type, and the second, of the two-step free nilpotent Lie groups (denoted $N_{v,2}$ for $v$ generators). In the latter case, we develop a radial Fourier calculus. Our study has focused on the maximal functions associated with Korányi spheres, together with their square functions, and the convolution operator defined with the radial Fourier calculus on the two-step free nilpotent Lie group (radial Fourier multipliers problem). In fact, one chapter of this work is devoted to the proof of $L^p$-inequalities for the maximal spherical function on the two considered classes of groups. Our method is based on interpolation for the same operator family as in the euclidean case, on $L^p$-boundedness for the standard maximal function, and $L^2$-inequalities for square functions. These $L^2$-inequalities are based on Plancherel formula and on the properties of bounded spherical functions for the orthogonal group. On $N_{v,2}$, we construct the bounded spherical functions using representations of the semidirect product of $N_{v,2}$ with the orthogonal group. We also obtain some properties of the Kohn sublaplacian and the radial Plancherel measure. Then we present a first study of the radial Fourier multiplier problem, with the aim of giving our solutions for some technicals difficulties.
We consider here the free two step nilpotent Lie group, provided with the homogeneous Korányi norm; we prove the $L^p$-boundedness of the maximal function corresponding to the homogeneous unit sphere, for some $p$.
The spectrum of a Gelfand pair $(K\ltimes N, K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz $K$-invariant functions on $N$. We also show the converse in the case of the Gelfand pair $(SO(3)\ltimes N_{3,2}, SO(3))$, where $N_{3,2}$ is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.