Born-Jordan Pseudo-Differential Operators with Symbols in the Shubin Classes

We apply Shubin's theory of global symbol classes $\Gamma_{\rho}^{m}$ to the Born-Jordan pseudodifferential calculus we have previously developed. This approach has many conceptual advantages, and makes the relationship between the conflicting Born-Jordan and Weyl quantization methods much more limpid. We give, in particular, precise asymptotic expansions of symbols allowing to pass from Born-Jordan quantization to Weyl quantization, and vice-versa. In addition we state and prove some regularity and global hypoellipticity results.


Introduction
The Born-Jordan quantization rules [5,6,8] have recently been rediscovered in mathematics and have quickly become a very active area of research under the impetus of scientists working in signal theory and time-frequency analysis [1,2,9,12]. It has been realized, not only that the associated phase space picture has many advantages compared with the usual Weyl-Wigner picture (it allows a strong damping of unwanted interference patterns [1,11]), but also, as one of us has shown [19,20,21], that there is strong evidence that Born-Jordan quantization might very well be the correct quantization method in quantum physics. Independently of these potential applications, the Born-Jordan pseudo-differential calculus has many interesting and difficult features (some of them, as non-injectivity [10], being even quite surprising) and deserve close attention. The involved mathematics is less straightforward than that of the usual Weyl formalism; for instance Born-Jordan pseudo-differential calculus is not fully covariant under linear symplectic transformations [17], which makes the study of the symmetries of the operators much less straightforward than in the Weyl case.
In the present paper we set out to study the pseudo-differential calculus associated with Born-Jordan quantization in the framework of Shubin's [29] global symbol classes. These results complement and extend those obtained by us in [10].
To be precise, in the Weyl quantization scheme to any observable (symbol) a(z), z ∈ R 2n , defined as a function or (temperate) distribution in phase space, it is associated the Weyl operator where a σ = F σ a is the symplectic Fourier transform of a and T (z 0 ) is the Heisenberg operator given by This is simply a phase space shift and, as a consequence of the Schwartz kernel theorem, every continuous linear operator S(R n ) → S ′ (R n ) can be written in a unique way as a Weyl operator for a suitable symbol a ∈ S ′ (R 2n ); namely, it is a superposition of phase space shifts. In this functional framework the Weyl correspondence between observables and operators is therefore one to one.
The Born-Jordan quantization of a symbol a(z) is instead defined as with z = (x, p) and px = p · x. The presence of the function sinc px 2 and in particular its zeros make the corresponding quantization problem much more subtle. It was proved in [10] that every linear continuous operator S(R n ) → S ′ (R n ) can still be written in Born-Jordan form, but the representation is no longer unique. The Born-Jordan correspondence is anyway still surjective.
In this paper we continue this investigation by focusing on a particulary relevant subclass of smooth symbols satisfying good growth conditions at infinity, namely Shubin's classes [29]. Roughly speaking the main result reads as follows. Within such symbol classes the Weyl symbol a W and the corresponding Born-Jordan symbol a BJ are related by the following explicit asymptotic expansions: and for suitable coefficients c α (see (4.10) below). These expansions seem remarkable, because at present there is no an exact and explicit formula for the Born-Jordan symbol corresponding to a given Weyl operator, although the existence of such a symbol was proved in [10]. Indeed, the situation seems definitely similar to what happens in the division problem of temperate distributions by a (not identically zero) polynomial P : the map f → P f from S ′ (R n ) into itself is onto but in general does not exist a linear continuous right inverse [4,26].
We refer to [30] for an alternative formulation of quantum observables by formal series of noncommutative generators x, p.
We will systematically use properties of the global pseudo-differential calculus whose study was initiated by Shubin, after related work by Beals, Berezin, Kumano-go, Rabinovič, and others (see the Bibliography in [29]). This calculus plays an important role in quantum mechanics since the position and momentum variables are placed on an equal footing in the estimates defining the symbol classes. We have found this approach particularly well adapted to investigate asymptotic expansions such as those for a W and a BJ .
Natural related topics that we have not included in this work are the spectral theory of Born-Jordan operators, in which the notion of global hypoellipticity plays a crucial role and the anti-Wick version of these operators (the latter might lead to some new insights). Finally, we have not discussed at all the Wigner-Moyal formalism associated with Born-Jordan question; for the latter we refer to [1,11,19].
In short, the paper is organized as follows. In Section 2 we review the definition of the Born-Jordan pseudodifferential operators. Section 3 is devoted to Shubin's symbol classes. In Section 4 we prove the above relationships between Weyl and Born-Jordan symbol. Finally Section 5 is devoted to applications to the global regularity problem.
Notation. We denote by σ the standard symplectic form n j=1 dp j ∧ dx j on the phase space R 2n ≡ R n × R n ; the phase space variable is written z = (x, p). Equivalently, σ(z, z ′ ) = Jz · z ′ where J = 0 I −I 0 . We will denote by x j the operator of multiplication by x j and set p j = −i ∂/∂x j . These operators satisfy Born's canonical commutation relations [ x j , p j ] = i where is a positive parameter such that 0 < ≤ 1.

Born-Jordan pseudo-differential Operators
In this section we review the recent advances in the theory of Born-Jordan quantization; for proofs and details we refer to Cordero et al. [10], de Gosson [17,19,20].
2.1. The Born-Jordan quantization rules. Following Heisenberg's insightful work on "matrix mechanics" Born and Jordan [5,6] proposed the quantization rule for monomials. Their rule conflicts with Weyl's [32] quantization rule, leading to (McCoy rule [28]) as soon as r ≥ 2 and s ≥ 2. The following observation is crucial: both quantizations are obtained from Shubin's τ -rule but by very different means. In fact, the Weyl rule (2.2) is directly obtained by choosing τ = 1 2 while Born and Jordan's rule (2.1) is obtained by averaging the right-hand side of (2.3) with respect to τ over the interval [0, 1] (de Gosson and Luef [22], de Gosson [17,19]).
On the operator level, the Weyl operator A W = Op W (a) is given by the familiar formula due to Weyl himself [32] (2.4) where a σ = F σ a is the symplectic Fourier transform and T (z 0 ) = e − i ℏ σ(ẑ,z 0 ) is the Heisenberg operator; recall [15,27] that the action of T (z 0 ) on a function or distribution ψ is explicitly given by Let us underline that the parameter ℏ ∈ (0, 1] is fixed in our context. Here we are not interested in the semi-classical analysis, i.e. the asymptotic as ℏ → 0. Using Plancherel's identity formula (2.4) can be rewritten is the Grossmann-Royer reflection operator (where Πψ(x) = ψ(−x)). One verifies that under suitable convergence conditions (for instance a ∈ S(R 2n ) and ψ ∈ L 1 (R n )) one recovers the more familiar "mid-point formula" 2π n e i p(x−y) a( 1 2 (x + y), p)ψ(y)d n yd n p common in the theory of pseudo-differential operators; we will use this notation as a formal tool for the sake of clarity (keeping in mind that it can be given a rigorous meaning by (2.7)). The easiest way to define Shubin's τ -operator A τ = Op τ (a) is to use the formula above as a starting point, and to replace the midpoint 1 2 (x + y) with (1 − τ )x + τ y which leads to As in the monomial case, the Born-Jordan operator A BJ = Op BJ (a) is obtained by averaging (2.10) over [0, 1]:

2.2.
Harmonic representation of Born-Jordan operators. The following result gives an explicit expression of the Weyl symbol of a Born-Jordan operator with arbitrary symbol (see [1,10]): (ii) The restriction of A BJ to monomials p s j x r j is given by the Born-Jordan rule (2.1).
Recall that the function sinc is defined by sinc u = sin u/u for u = 0 and sinc 0 = 1.
3.1. The Shubin symbol class Γ m ρ . We begin by giving the following definition (Shubin [29], Definition 23.1): ; using Leibniz's rule for the derivative of products of functions one easily checks that The class Γ m ρ (R 2n ) is a complex vector space for the usual operations of addition and multiplication by complex numbers, and we have The reduced harmonic oscillator Hamiltonian H(z) = 1 2 (|x| 2 + |p| 2 ) obviously belongs to Γ 2 1 (R 2n ), and so does in fact, any polynomial function in z of degree m is in Γ m 1 (R 2n ). In particular every Hamiltonian function of the type The following lemma shows that the symbol classes Γ m ρ (R 2n ) are invariant under linear automorphisms of phase space (this property does not hold for the usual Hörmander classes S m ρ,δ (R n ) [25], whose elements are characterized by growth properties in only the variable p). Let us denote by GL(2n, R) the space of 2n × 2n invertible real matrices. Then Proof. The result is showed in greater generality in [29, p. 177]. This special case simply follows by the fact for a suitable C > 0.

3.2.
Asymptotic expansions of symbols. Let us recall the notion of asymptotic expansion of a symbol a ∈ Γ m ρ (R 2n ) (cf. [29], Definition 23.2): where m r = max j≥r m j we will write a ∼ ∞ j=1 a j and call this relation an asymptotic expansion of the symbol a.
The interest of the asymptotic expansion comes from the fact that every sequence of symbols (a j ) j with a j ∈ Γ m j ρ (R 2n ), the degrees m j being strictly decreasing and such that m j → −∞, determines a symbol in some Γ m ρ (R 2n ), that symbol being unique up to an element of S(R 2n ): (See Shubin [29], Proposition 23.1). Note that property (ii) immediately follows from (3.3).
3.3. The amplitude classes Π m ρ . We will need for technical reasons an extension of the Shubin classes Γ m ρ (R 2n ) defined above. Since Born-Jordan operators are obtained by averaging Shubin's τ -operators 1] we are led to consider pseudo-differential operators of the type is called amplitude and is defined, not on R 2n ≡ R n x × R n p but rather on R 3n ≡ R n x × R n y × R n p . It therefore makes sense to define an amplitude class generalizing Γ m ρ (R 2n ) by allowing a dependence on the three sets of variables x, y, and p (cf. [29] Definition 23.3): Definition 3. Let m ∈ R. The symbol (or amplitude) class Π m ρ (R 3n ) consists of all functions a ∈ C ∞ (R 3n ) such that for some m ′ ∈ R satisfy It turns out that an operator (3.5) with amplitude b ∈ Π m ρ (R 3n ) is a Shubin τ -pseudo-differential operator with symbol in Γ m ρ (R 2n ) -and this for every value of the parameter τ : Proposition 3. Let τ be an arbitrary real number. (i) Every pseudodifferential operator A of the type (3.5) with amplitude b ∈ Π m ρ (R 3n ) can be uniquely written in the form A = Op τ (a τ ) for some symbol a τ ∈ Γ m ρ (R 2n ), that is the symbol a τ has the asymptotic expansion Proof. See Shubin [29], Theorem 23.2 for the case = 1 and de Gosson [16], Section 14.2.2.
We have in addition an asymptotic formula allowing to pass from one τ -symbol to another when A is given by ( It is useful to make the following remark: in the standard theory of pseudo-differential operators (notably in its applications to partial differential operators) it is customary to use operators (3.9) Aψ(x) = 1 2π n e i(x−y)ξ a(x, y, ξ)ψ(y)d n yd n ξ which correspond, replacing p with ξ to the choice = 1 in the expression (3.5). It is in fact easy to toggle between the expression above and itsdependent version: one just replaces a(x, y, ξ) with a(x, y, p) and ξ with p/ so that dξ = −n dp. However, when doing this, one must be careful to check that the amplitude a(x, y, ξ) and a(x, y, ξ) belong to the same symbol class. That this is indeed always the case when one deals with Shubin classes is clear from Lemma 1. The following situation is important in our context; consider the = 1 Weyl operator (3.10) Aψ(x) = 1 2π n e i(x−y)ξ a( 1 2 (x + y), ξ)ψ(y)d n yd n ξ.
Denoting by A ( ) the corresponding operator (3.5) in order to make the -dependence clear, that is Using the symbol estimates (3.1) it is straightforward to show that every operator A ∈ G m ρ (R n ) is a continuous operator S(R n ) −→ S(R n ) and can hence be extended into a continuous operator S ′ (R n ) −→ S ′ (R n ). It follows by duality that if A ∈ G m ρ (R n ), then A * ∈ G m ρ (R n ) (cf. [29], Theorem 23.5). One also shows that ( [29,Theorem 23.6 The difficulty in recovering b σ from a σ comes from the fact that the sinc function has infinitely many zeroes; in fact sinc(px/2 ) = 0 for all points z = (x, p) such that px = 2N π for a non-zero integer N . We are thus confronted with a division problem. Notice in addition that if the solution b exists then it is not unique: assume that c(z) = e −iσ(z,z 0 )/ where p 0 x 0 = 2N π (N ∈ Z, N = 0). We have c σ (z) = (2π ) n δ(z − z 0 ) and hence by (2.14) It follows that if Op BJ (b) = Op W (a) then we also have Op BJ (b + c) = Op W (a). Now, in [10, Theorem 7] we have proven that the equation (4.1) always has a (non-unique) solution in b ∈ S(R 2n ) for every given a ∈ S(R 2n ); our proof used the theory of division of distributions. Thus every Weyl operator has a Born-Jordan symbol; equivalently

Proposition 4. For every continuous linear operator
Notice that the existence of the solution b of (4.1), as established in [10], is a purely qualitative result; it does not tell us anything on the properties of that solution.

4.2.
Weyl symbol of a Born-Jordan operator. We are going to show that every Born-Jordan operator with symbol in one of the Shubin classes Γ m ρ (R 2n ) is a Weyl operator with symbol in the same symbol class and produce an asymptotic expansion for the latter. For this we will need the following elementary inequalities: Lemma 2. Let ξ and η be positive numbers and m ∈ R. We have Proof. Proof of (4.2). The case m ≥ 0 is straightforward: we have the case ξ > η follows in the same way. Proof of (4.3): see for instance Chazarain and Piriou [7] or Hörmander [25].
The estimate (4.3) is usually referred to as Peetre's inequality in the literature on pseudo-differential operators.
Here a τ has the following asymptotic expansion , having the asymptotic expansion and we have a W − a ∈ Γ m−2ρ ρ (R 2n ).
Notice that the asymptotic formula (4.5) yields exact results when the Born-Jordan symbol a is a polynomial in the variables x j , p k . For instance, when n = 1 and a(z) = a rs (z) = x r p s it leads to (4.9) a rs,W (x, p) = k≤inf(r,s) k even i 2 We refer to Domingo and Galapon [23] for a general discussion of quantization of monomials. Using [29], Definition 23.4, the result above has the following interesting consequence: This result reduces in many cases the study of Born-Jordan operators to that of Shubin operators.
4.3. The Born-Jordan symbol of a Weyl operator. We now address the more difficult problem of finding the Born-Jordan symbol of a given Weyl operator in G m ρ (R 2n ). As already observed the analysis in [10] did not provide an explicit formula for it because of division problems. It is remarkable that, nevertheless, an explicit and general asymptotic expansion can be written down when the symbol belongs to one of the classes Γ m ρ (R 2n ). To this end we need a preliminary lemma about the formal power series arising in (4.5).
Its formal reciprocal is given by the series α∈N n cα α! x α where c 0 = 1 and, for α = 0, (4.10) Proof. The proof is straightforward: we expand 1 + |α| =0 even 1 α!(|α| + 1) x α −1 as a geometric series and collect the similar terms. Alternatively, we could also apply the Faà di Bruno formula generalizing the chain rule to the derivatives at x = 0 of the function x −→ g(f (x)) where g(t) = 1/(1 + t) and Let us now prove our second main result: ) be any symbol (whose existence is guaranteed by Proposition 2) with the following asymptotic expansion: where the coefficients c α are given in (4.10) (c 0 = 1).
Let A BJ = Op BJ (b) be the corresponding Born-Jordan operator. Then (4.12) where R is a pseudodifferential operator with symbol in the Schwartz space S(R 2n ).
Proof. The operator A BJ = Op BJ (b) by Theorem 1 can be written as a Weyl operator with Weyl symbol .
Now we substitute in this expression the asymptotic expansion (4.11) for b and we use the fact that the formal differential operators given by the series α∈N n |α| even are inverses of each other in view of Lemma 3 (to see this, formally replace x = (x 1 , . . . , x n ) in Lemma 3 by (i /2)(∂ x 1 ∂ p 1 , . . . , ∂ xn ∂ pn ). It follows that hence (4.12).
Notice that in dimension n = 1 we have k≥0 even 1 k!(k + 1) x k = 1 x k≥0 even so that c k = ∂ k x (1/F (x))| x=0 . In particular, c k = 0 for odd k. In this case the series expansion of 1/F (x) is particularly easy, since it coincides with the MacLaurin series expansion of the function where the B k are the Bernoulli numbers More explicitly, 1 x 4 − 31 15120 x 6 + 127 604800 x 8 − · · · and the coefficients c k , with k even, are provided by In this case formula (4.11) takes the simple form As in the case of formula (4.9), the asymptotic expansion (4.11) becomes exact (and reduces to a finite sum) when the symbol a is a polynomial. For instance, assuming n = 1 choose a(z) = a rs (z) = x r p s . Then the formula above yields b rs,BJ (x, p) = k≤inf(r,s) k even where the B k are the Bernoulli numbers defined in (4.15). We also make the following remark: formulas (4.13) and (4.14) show that (modulo a term in S(R 2n )) a Weyl operator with symbol in Γ m ρ (R 2n ) has a Born-Jordan symbol belonging to the same class Γ m ρ (R 2n ). This is however by no means a uniqueness result since, as we have already observed, we have Op BJ (b + c) = 0 for all symbols c(z) = e −iσ(z,z 0 )/ where p 0 x 0 = 2N π (N ∈ Z, N = 0). Observe that such a symbol c belongs to none of the symbol classes Γ m ρ (R 2n ).

Regularity and Global Hypoellipticity Results
In order to define the Sobolev-Shubin spaces (cf. [29], Definition 25.3), we recall the definition of anti-Wick operators. The anti-Wick operator Op AW (a) with symbol a is defined by where P z f (t) = f, Φ z Φ z (t) are orthogonal projections on L 2 (R n ) on the functions Φ z (t) = π −n/4 e itp e − |t−x| 2 2 , z = (x, p) ∈ R 2n (i.e., phase-space shifts of the Gaussian π −n/4 e − |t| 2 2 ).
Definition 4. For s ∈ R consider the anti-Wick symbol z s , z ∈ R 2n , and let A s = Op AW (a) be the corresponding anti-Wick operator. The Sobolev-Shubin space Q s is defined by . We then have the following continuity result for Born-Jordan operators: Proof. By Corollary (1) the operator A BJ is in the class G m ρ (R n ). The result follows by applying Theorem 25.2 in [29].
It turns out that the Sobolev-Shubin spaces are particular cases of Feichtinger's modulation spaces [13,14,24]; we do not discuss these here and refer to Cordero et al [12] for a study of continuity properties of Born-Jordan operators in these spaces. In fact, our reduction result Theorem 1 allows to transpose to Born-Jordan operators all known regularity results for Weyl operators with symbol in the symbol classes Γ m ρ (R 2n ). For instance: Proposition 6. Let a ∈ Γ 0 ρ (R 2n ). Then A BJ = Op BJ (a) is a bounded operator on L 2 (R n ).
Proof. In view of Theorem 24.3 in [29] every Weyl operator with symbol in Γ 0 ρ (R 2n ) is bounded on L 2 (R n ); the result follows in view of Theorem 1 (ii).
We next recall the notion of global hypoellipticity [29,31], which plays an important role in the study of spectral theory for pseudo-differential operators (see the monograph [3] by Boggiatto et al.). An operator A : S ′ (R n ) −→ S ′ (R n ) which also maps S(R n ) into itself is globally hypoelliptic if ψ ∈ S ′ (R n ) and Aψ ∈ S(R n ) =⇒ ψ ∈ S(R n ).
(global hypoellipticity is thus not directly related to usual notion of hypoellipticity [25], which is a local notion).
In [29] Shubin introduces the following subclass of Γ m ρ (R 2n ): Definition 5. Let m, m 0 ∈ R and 0 < ρ ≤ 1. The symbol class HΓ m,m 0 ρ (R 2n ) consists of all complex functions a ∈ C ∞ (R 2n ) such that C 0 z m 0 ≤ |a(z)| ≤ C 1 z m for |z| > R, for some C 0 , C 1 , R > 0 and whose derivatives satisfy the following property: for every α ∈ N 2n there exists C α > 0 such that The interest of these symbol classes comes from the following property [29]: if a ∈ HΓ m,m 0 ρ (R 2n ) then the Weyl operator A = Op W (a) is globally hypoelliptic. For instance, the Hermite operator −∆ + |x| 2 is globally hypoelliptic since its Weyl symbol is a(z) = |z| 2 , which is in HΓ 2,2 1 (R 2n ).
Proposition 7. Let a ∈ HΓ m,m 0 ρ (R 2n ), with m − 2ρ < m 0 . (i) The Born-Jordan operator A BJ = Op BJ (a) is globally hypoelliptic; (ii) if ψ is a tempered distribution such that A BJ ψ ∈ Q s (R n ) for some s ∈ R then ψ ∈ Q s+m 0 (R n ).
Proof. In view of the discussion above it suffices to show that the Weyl symbol a W of A BJ belongs to the class HΓ m,m 0 ρ (R 2n ). Now, by Theorem 1 (ii) we have a W − a ∈ Γ m−2ρ ρ (R 2n ) hence the result follows using of Lemma 4.