On Sobolev norms for Lie group representations
Introduction
Let G be a Lie group and be a Banach representation of G, that is, a morphism of groups such that the orbit maps are continuous for all .
We say that a vector v is k-times differentiable if and write for the corresponding subspace. The smooth vectors are then defined by .
The space 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 is defined as follows: Here is a fixed basis for the Lie algebra of G, and is, as usual, the derived representation for the universal enveloping algebra of . Then , endowed with the norm , is a Banach space and defines a Banach representation of G. Furthermore, carries a natural Fréchet structure, induced by the Sobolev norms . The corresponding G-action on is smooth and of moderate growth, i.e. an SF-representation in the terminology of [2].
In case is a unitary representation on a Hilbert space , there is an efficient way to define the Fréchet structure on via a Laplace element in . More specifically, one defines the 2k-th Laplace Sobolev norm in this case by The unitarity of the action then implies that the standard Sobolev norm is equivalent to .
For a general Banach representation we still have , but it is no longer true that , as defined in (1.3), is equivalent to : it typically fails that is dominated by , for example if 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 . 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 admits a constant such that the operator 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 .
More precisely, for the Laplace Sobolev norms defined as we show that the families and are equivalent in the following sense: Let m be the smallest even integer greater or equal to . Then there exist constants such that
The above mentioned results follow from a novel factorization of the delta distribution 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 , 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 by On the other hand [2] provided another definition of Sobolev norms for any order ; they were denoted and termed induced Sobolev norms there. The norms were based on a noncanonical localization to a neighborhood of , identified with the unit ball in , and used the s-Sobolev norm on . We show that the two notions and are equivalent up to constant shift in the parameter s, see Proposition 4.3. The more geometrically defined norms may therefore replace the norms 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 and in this context it is often quite cumbersome to estimate the dual norm of , caused by the many terms in the definition (1.1). On the other hand the dual norm of , as defined by one operator , 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 the distance function associated to g (i.e. the infimum of the lengths of all paths connecting group elements g and h), by the ball of radius r centered at g, and we set Here are two key properties of , which will be relevant later, see [5]: Lemma 2.1 If 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 .
For a Banach space E we denote by the associated group of topological linear isomorphisms. By a Banach representation of a Lie group G we understand a group homomorphism such that the action is continuous. For a vector we denote by the corresponding continuous orbit map. Given , the
Standard and Laplace Sobolev norms
As before, we let be a Banach representation. On , the space of smooth vectors, one usually defines Sobolev norms as follows. Let p be the norm underlying E. We fix a basis of and set Strictly speaking this notion depends on the choice of the basis and would be the more accurate notation. However, a different choice of basis, say leads to an equivalent family of norms , i.e. for all k there exist
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