In this section we discuss various aspects of the Heisenberg group, hopefully shedding light on its importance and general structure. We remind the reader that the Heisenberg groupHn in Example 1.6.4 was defined in the following way: the Heisenberg groupHis the manifold R2n+1 endowed with the law. Although we will not use them, let us mention some other important appearances of the Heisenberg group.
The Heisenberg group can also be realized as a group of transformations; for example for each. Therefore, the Heisenberg group Hn described above in Section 6.1.1, i.e. R2n+1, equipped with the group law given in (6.1), is a connected simply connected (nilpotent degree-two) Lie group whose Lie algebra is hn and which is realized via an exponential mapping together with a canonical basis. The Heisenberg Lie algebra is stratified viahn =V1⊕V2, where V1 is linearly spanned by Xj and Yj while V2=RT.
Since the Heisenberg Lie algebra is stratified viahn=V1⊕V2, the natural dilations on the Lie algebra are given by. The Schrödinger representations of the Heisenberg group Hn are the infinite-dimensional unitary representations of Hn, allowing ourselves, as usual, to identify unitary representations by their unitary equivalence classes. The Schwartz space on the Heisenberg group Hn, realized as we have done, is defined as S(R2n+1), see section 3.1.9.
The characterization of the Fourier image of a (full) Schwartz space on Hn is a difficult problem analyzed by Geller in [Gel80].
Plancherel measure
Since the integral kernel off(πλ) is square integrable, the operatorif(πλ) is Hilbert-Schmidt and its Hilbert-Schmidt norm is the L2 norm of its integral kernel (see, e.g., [RS80, Theorem VI .23]). From Plancherel's formula in proposition 6.2.7 it follows that the Schr¨o-dinger representations πλ, λ ∈ R\{0}, are almost all representations of Hn.
Difference operators
Difference operators Δ x j and Δ y j
Although we could just use direct calculations, we prefer to use the following observations. Above and also below we use the formula for the symbols of real derivatives, for example,πλ( ˜Yjκ) =πλ(κ)πλ(Yj), see Proposition 1.7.6, (iv). Ifκ(πλ) = OpW(aλ) andaλ={aλ(ξ, u)}as in the statement above, we will often say ataλ is the λ symbol.
So far we have analyzed the differentiators used for the ‘general’ group Fourier transform of the κ distribution (provided the differentiators were meaningful, see Definition 5.2.1 and further discussion). In the following, we specifically classify this to some known symbols, especially to the one in example 5.1.26, i.e. to π(A), where A is a left-invariant difference operator, such as A=Xj, Yj or T.
Difference operator Δ t
Before we give some examples of applications of the difference operator Δt, we want to make a few remarks. Similar to Note 6.3.2, the formula in Lemma 6.3.6 respects the properties of the automorphism Θ and the dilations Dr. Now considering Xj and Yj as elements of the Lie algebra and invariant vector fields, we see that we use (6.23) and (6.6).
With the help of formulas (6.28), Corollary 6.3.3 and the properties of the Weyl calculus (see especially the compound formula in (6.16)), we easily obtain this.
Formulae
Shubin classes
- Weyl-H¨ ormander calculus
- Shubin classes Σ m ρ (R n ) and the harmonic oscillator
- Shubin Sobolev spaces
- The λ -Shubin classes Σ m ρ,λ (R n )
- Commutator characterisation of λ -Shubin classes
The symbol class S(M, g) is the set of functions a∈C∞(R2n) such that for each integer∈N0, the quantity. Furthermore, there exists a structural constant C >0 and a structural integer∈N0 such that for any symbol we have a∈S(1, g). The following is well known and can be considered more generally as a consequence of the Weyl-H¨ormander calculus (see Theorem 6.4.9).
For each s ∈ R, Qs(Rn) coincides with the complement (in S(Rn)) of the Schwartz spaceS(Rn) for the norm. From the duality of the complex interpolation and of Qs(Rn), we obtain the inverse inclusion and part (7) is proved. Of the calculus properties it is again a routine exercise left to the reader that the dual of Q(sb)(Rn) is Q(−sb)(Rn) via distributive duality and that the spaces Q(sb)(Rn) decrease with ∈R.
With the duality of complex interpolation and Qs(Rn) spaces, we obtain the inverse inclusion and (6.49) is proved. Let us show that for each ∈N0 the space Qs(Rn) coincides with the space Q(sint)(Rn) of the functions sh∈ L2(Rn), so that the tempered distributions uα∂uβh in L2(Rn) for each α , β ∈ Nn0 such that |α|+|β| holds ≤ s. Moreover, for any ∈N there exists a constant C and an integer independent of A such that.
The operator A is inΨΣmρ(Rn)if and only if γo ∈R exists so that for every α1 we have α2∈Nn0. To show that the metric gρ,λ varies slowly, we note that it has the form φ(X)−2|T|2 as in Example 6.4.5 with. Here it means that Σmρ,λ(Rn) is the class of functions a∈ C∞(Rn×Rn), so that for eachN∈N0 is the quantity.
It is clear that all spaces of the same order and parameterρ coincide in the sense that. In particular, the constant operator πλ(T) = iλ should be considered to be of order 2 because of its dependence on λ. First we need to understand some properties of the Sobolev spaces related to the λ-dependent metric used to define the λ-Shubin symbols.
Moreover, for any∈N there exists a constant C and an integer, both independent of {Aλ} and λ, such that. Furthermore, for each γ∈Rand∈N there exists a constant C and an integer , both independent of {Aλ} and λ, such that.
Quantization on the Heisenberg group
