On Sobolev norms for Lie group representations

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Abstract

We define Sobolev norms of arbitrary real order for a Banach representation (π,E) of a Lie group, with regard to a single differential operator D=dπ(R2+Δ). Here, Δ is a Laplace element in the universal enveloping algebra, and R>0 depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for D on the space of smooth vectors of E. The main tool is a novel factorization of the delta distribution on a Lie group.

Introduction

Let G be a Lie group and (π,E) be a Banach representation of G, that is, a morphism of groups π:GGL(E) such that the orbit mapsγv:GE,gπ(g)v, are continuous for all vE.

We say that a vector v is k-times differentiable if γvCk(G,E) and write EkE for the corresponding subspace. The smooth vectors are then defined by E=k=0Ek.

The space Ek carries a natural Banach structure: if p is a defining norm for the Banach structure on E, then a k-th Sobolev norm of p on Ek is defined as follows:pk(v):=(m1++mnkp(dπ(X1m1Xnmn)v)2)12(vEk). Here X1,,Xn is a fixed basis for the Lie algebra g of G, and dπ:U(g)End(E) is, as usual, the derived representation for the universal enveloping algebra U(g) of g. Then Ek, endowed with the norm pk, is a Banach space and defines a Banach representation of G. Furthermore, E carries a natural Fréchet structure, induced by the Sobolev norms (pk)kN0. The corresponding G-action on E is smooth and of moderate growth, i.e. an SF-representation in the terminology of [2].

In case (π,H) is a unitary representation on a Hilbert space H, there is an efficient way to define the Fréchet structure on H via a Laplace elementΔ=j=1nXj2 in U(g). More specifically, one defines the 2k-th Laplace Sobolev norm in this case byp2kΔ(v):=p(dπ((1+Δ)k)v)(vE2k). The unitarity of the action then implies that the standard Sobolev norm p2k is equivalent to p2kΔ.

For a general Banach representation (π,E) we still have E=k=0dom(dπ(Δk)), but it is no longer true that p2kΔ, as defined in (1.3), is equivalent to p2k: it typically fails that p2k is dominated by p2kΔ, for example if 1spec(dπ(Δ)) or if elliptic regularity fails as in Remark 4.2 below.

In the following we use Δ for the expression (2.1), a first-order modification of Δ as defined in (1.2), in order to make Δ selfadjoint on L2(G). In case G is unimodular, we remark that the two notions (2.1) and (1.2) coincide.

One of the main results of this note is that every Banach representation (π,E) admits a constant R=R(E)>0 such that the operator dπ(R2+Δ):EE is invertible, see Corollary 3.3. The constant R is closely related to the growth rate of the representation, i.e. the growth of the weight wπ(g)=π(g).

More precisely, for the Laplace Sobolev norms defined asp2kΔ(v):=p(dπ((R2+Δ)k)v)(vE2k), we show that the families (p2k)k and (p2kΔ)k are equivalent in the following sense: Let m be the smallest even integer greater or equal to 1+dimG. Then there exist constants ck,Ck>0 such thatckΔp2k(v)p2k(v)CkΔp2k+m(v)(vE).

The above mentioned results follow from a novel factorization of the delta distribution δ1 on G, see Proposition 2.4 in the main text for the more technical statement. This in turn is a consequence of the functional calculus for Δ, developed in [3], and previously applied to representation theory in [7] to derive factorization results for analytic vectors. The functional calculus allows us to define Laplace Sobolev norms for any order sR bypsΔ(v):=p(dπ((R2+Δ)s2)v)(vE). On the other hand [2] provided another definition of Sobolev norms for any order sR; they were denoted Sps and termed induced Sobolev norms there. The norms Sps were based on a noncanonical localization to a neighborhood of 1G, identified with the unit ball in Rn, and used the s-Sobolev norm on Rn. We show that the two notions psΔ and Sps are equivalent up to constant shift in the parameter s, see Proposition 4.3. The more geometrically defined norms psΔ may therefore replace the norms Sps in [2].

Our motivation for this note stems from harmonic analysis on homogeneous spaces, see for example [1] and [4]. Here one encounters naturally the dual representation of some Ek and in this context it is often quite cumbersome to estimate the dual norm of pk, caused by the many terms in the definition (1.1). On the other hand the dual norm of psΔ, as defined by one operator dπ((R2+Δ)s2), is easy to control and simplifies a variety of technical issues.

Section snippets

Some geometric analysis on Lie groups

Let G be a Lie group of dimension n and g a left invariant Riemannian metric on G. The Riemannian measure dg is a left invariant Haar measure on G. We denote by d(g,h) the distance function associated to g (i.e. the infimum of the lengths of all paths connecting group elements g and h), by Br(g)={xG|d(x,g)<r} the ball of radius r centered at g, and we setd(g):=d(g,1)(gG). Here are two key properties of d(g), which will be relevant later, see [5]:

Lemma 2.1

If w:GR+ is locally bounded and

Banach representation of Lie groups

In this section we briefly recall some basics on Banach representation of Lie groups and apply Proposition 2.4 to the factorization of vectors in Ek.

For a Banach space E we denote by GL(E) the associated group of topological linear isomorphisms. By a Banach representation (π,E) of a Lie group G we understand a group homomorphism π:GGL(E) such that the actionG×EE,(g,v)π(g)v, is continuous. For a vector vE we denote byγv:GE,gπ(g)v, the corresponding continuous orbit map. Given kN0, the

Standard and Laplace Sobolev norms

As before, we let (π,E) be a Banach representation. On E, the space of smooth vectors, one usually defines Sobolev norms as follows. Let p be the norm underlying E. We fix a basis B={X1,,Xn} of g and setpk(v):=[m1++mnkp(dπ(X1m1Xnmn)v)2]12(vE). Strictly speaking this notion depends on the choice of the basis B and pk,B would be the more accurate notation. However, a different choice of basis, say C={Y1,,Yn} leads to an equivalent family of norms pk,C, i.e. for all k there exist

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