A pseudo-differential calculus on the Heisenberg group

In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,3,2] of pseudo-differential calculi on graded groups. The relation between the Weyl quantization and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.


Introduction
In [4], see also [3,2], a pseudo-differential calculus is developed in the setting of graded Lie groups using their representations. Here we present the results of this construction in the particular case of the Heisenberg group H n .
It is well known that the representations of H n are intimately linked with the Weyl quantization on R n (see e.g., [10], and Section 2 below). Together with the analogue of the Kohn-Nirenberg quantization on Lie groups (see e.g., [10,7,4], and Section 4 below), this link enables the development of pseudo-differential calculi on H n with scalar-valued symbols that depend on parameters. However, the remaining difficulty lies in finding conditions to be imposed on those symbols so that the resulting class of operators has the expected properties of a calculus.
Although M. Taylor explained these general ideas in the setting of the Heisenberg groups in [10], he chose to restrict his analysis in [10] mainly to invariant (i.e. convolution) operators on H n with symbols defined by some asymptotic expansions. To the authors' knowledge, the only study of non-invariant calculi with scalar-valued symbols on H n was done, until now, by H. Bahouri, C. Fermanian-Kammerer and I. Gallagher in [1]. Their work is devoted to the case of H n only. Moreover, the conditions imposed on the scalar-valued symbols might appear difficult to apprehend for some readers, as they come from technical parts of the proofs of the calculi's properties (see the more recent version of [1] on the server Hal). Our conditions on symbol classes differ from those in [1] for small λ. At the end of this note we list several applications of the analysis in our classes, to the hypoellipticity properties and subelliptic estimates for several operators on the Heisenberg group.
Our approach to find the conditions on the symbols is different from [10] and [1]: we particularise to the setting of H n our definition of pseudo-differential calculi valid on a large class of nilpotent Lie groups, namely the graded groups, see [4,3,2]. In our general construction, the symbols are operator-valued. Nonetheless on H n , using the link between the Weyl quantization and the representations of H n , this is equivalent to using the scalarvalued symbols. The purpose of this note is to present what the general conditions on the symbols given in [4,3,2] become when expressed on the level of scalar-valued symbols of H n . In particular, we find conditions which are similar but different to the ones in [1]. As applications for our analysis, we give sufficient condition for Schwartz hypoellipticity and for subelliptic estimates on the Heisenberg group.

Schrödinger representations and Weyl quantization
We start by fixing the notation required for presenting our results. We realise the Heisenberg group H n as the manifold R 2n+1 endowed with the law (x, y, t)(x , y , t ) = (x + x , y + y , t + t + 1 2 (xy − x y)), where (x, y, t) and (x y , t ) are in R n × R n × R = R 2n+1 . Here we adopt the convention that if x and y are two vectors in R n for some n ∈ N, then xy denotes their standard scalar product xy = n j=1 x j y j if x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ).
The canonical basis for the Lie algebra h n of H n is given by the left-invariant vector fields The canonical commutation relations are and T is the centre of h n . The Heisenberg Lie algebra is stratified via h n = V 1 ⊕ V 2 where V 1 is linearly spanned by the X j 's and Y j 's, while V 2 = RT . Therefore, the group H n is naturally equipped with the family of dilations D r given by D r (x, y, t) = r(x, y, t) = (rx, ry, r 2 t), (x, y, t) ∈ H n , r > 0.
The 'canonical' positive Rockland operator in this setting is R = −L, where L is the sub-Laplacian The Schrödinger representations of the Heisenberg group H n are the infinite dimensional unitary representations of H n (we allow ourselves to identify unitary representations with their unitary equivalence classes). Parametrised by λ ∈ R\{0}, they act on L 2 (R n ). We denote them by π λ and realise them as for h ∈ L 2 (R n ), u ∈ R n , and (x, y, t) ∈ H n , where we use the convention As already noted in [10], it can be effectively computed by for h ∈ L 2 (R n ) and u ∈ R n , that is, Here the Fourier transform and Op W denotes the Weyl quantization, which is given for a reasonable symbol a on R n ×R n , by where f ∈ S(R n ) and u ∈ R n . We keep the same notation π λ for the corresponding infinitesimal representations. We readily compute that With our choice of notation and definitions, the Plancherel measure is c n |λ| n dλ in the sense that the Plancherel formula holds for any κ ∈ S(H n ). For the value of the constant c n , see [4]. Here · HS denotes the Hilbert-Schmidt norm of an operator on L 2 (R n ), that is, B 2 HS = Tr(B * B). This allows one to extends unitarily the definition of the group Fourier transform to L 2 (H n ). Formula (4) then holds for any κ ∈ L 2 (H n ).

Difference operators
Difference operators were defined in [7,9] as acting on Fourier coefficients on compact Lie groups, and on graded nilpotent Lie groups in [4]. In the setting of the Heisenberg group, this yields the definition of the difference operators ∆ x j , ∆ y j , and ∆ t via ∆ x j κ(π λ ) := π λ (x j κ), ∆ y j κ(π λ ) := π λ (y j κ), ∆ t κ(π λ ) := π λ (tκ), for suitable distributions κ defined on H n . We can compute that When π λ (κ) = Op W (a λ ) and a λ = {a λ (ξ, u)}, we have For example, we have The following equalities shed some light on why, for example in [1], another normalisation of the Weyl symbol is preferred. Indeed, the expressions on the right-hand sides in (5), in particular for the operator∂ λ,ξ,u defined in (6), become very simple.

Quantization and symbol classes
In this note, for simplicity, we change slightly the notation with respect to the general case developed in [4]. Firstly we want to keep the letter x to denote part of the coordinates of the Heisenberg group and we choose to denote a general element of the Heisenberg group by, e.g., g = (x, y, t) ∈ H n .
Thirdly we modify the indices α ∈ N 2n+1 0 in order to write them as with α 1 = (α 1,1 , . . . , α 1,n ) ∈ N n 0 , α 2 = (α 2,1 , . . . , α 2,n ) ∈ N n 0 , α 3 ∈ N 0 . The homogeneous degree of α is then For each α we write . . x α 1n n , y α 2 = y α 21 1 . . . y α 2n n , and we define the corresponding difference operator We also write Following [4], we define the symbol class S m ρ,δ (H n ) as the set of symbols σ for which all the following quantities are finite: A natural quantization on any type I Lie group is the analogue of the Kohn-Nirenberg quantization on R n , see, e.g., [10] for general remarks, [7] for the consistent development in the case of compact Lie group, and [4] for the case of nilpotent Lie groups. In the particular case of the Heisenberg group, this quantization associates to a symbol σ (for example in S m ρ,δ (H n )) the operator A = Op(σ) acting on S(H n ) given by Tr (π λ (g)σ(g, λ)π λ (ϕ)) |λ| n dλ.
Here we have used our notation, especially for the Plancherel measure c n |λ| n dλ, see (4). We denote by the class of operators corresponding to the symbols in S m ρ,δ (H n ) via this quantization.
(2) Conversely, if a = {a(g, λ, ξ, u) = a g,λ (ξ, u)} is a smooth function on H n × R\{0} × R n × R n satisfying (10) for every α, β ∈ N n 0 ,α ∈ N 0 , then there exists a unique symbol σ ∈ S m ρ,δ (H n ) such that (9) holds. For the definition of the Sobolev spaces L 2 s (H n ) on H n and more generally on any stratified group, see [5]. Part (iii) summarises the main results of the general construction made in [4] on any graded groups.
For the value of the constant c n , see [4]. The formula (11) now involves mainly 'Euclidean objects'.
Corollary 5.1 has also a corresponding 'hypoellipticity version' which we omit here, but we give a few examples of both of them. First, let m, m o ∈ 2N be two even integers such that m ≥ m 0 . Let A be a differential operator given by either X m + iY mo + T mo/2 or X mo + iY m + T mo/2 on H 1 . Then A is Schwartz hypoelliptic and satisfies the subelliptic estimates ∀s ∈ R ∃C > 0 ∀f ∈ S(H 1 ) f L p s+mo (H 1 ) ≤ C Af L p s (H 1 ) . The above mentioned conclusion that I − L is Schwartz hypoelliptic can be also extended to variable coefficients using our calculus. For example, if f 1 and f 2 are complex-valued smooth functions on H n such that inf x∈Hn,λ≥Λ |f 1 (x) + f 2 (x)λ| 1 + λ > 0 for some Λ ≥ 0, and such that functions X α 1 f 1 , X α 2 f 2 are bounded for every α 1 , α 2 ∈ N n 0 , then the differential operator f 1 (x) − f 2 (x)L is Schwartz-hypoelliptic and satisfies the following subelliptic estimates ∀s ∈ R ∃C > 0 ∀ϕ ∈ S(H n ) ϕ L p s+2 (Hn) ≤ C f 1 ϕ − f 2 Lϕ L p s (Hn) .