On the Weyl transform for rotationally invariant symbols

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In Sect. 5, we present three more advanced examples: We characterize the boundedness and the compactness of the Weyl transforms for the symbols σ z = (x 2 + y 2 ) z , σ z = (x 2 +y 2 −ρ 2 ) z + , and σ p,z = χ(x 2 +y 2 )(x 2 +y 2 ) z exp(i(x 2 +y 2 ) p ), respectively, for z ∈ C\(−N), ρ > 0, p > 1 and χ ∈ C ∞ (R) with χ(t) = 1 for t ≥ 2, χ(t) = 0 for t ≤ 1, in dependence on the parameters z and p. (Note that the second example contains the Dirac measure on the circle as a limit case since lim z→−1 t z In these examples, the determination of the asymptotic behavior of the eigenvalues of the Weyl transforms plays an essential rôle. Let us introduce some notation. Besides the familiar Banach spaces L p (R n ), 1 ≤ p ≤ ∞, we use the Banach spaces of integrable measures M 1 (R n ), see [6, p. 345], and of bounded and of compact linear operators on L 2 (R n ), respectively, which we denote by L(L 2 (R n )) and by Com(L 2 (R n )), respectively, compare [11]. We employ the standard notation for the distribution spaces D , S , E , the dual spaces of the spaces D, S, E of "test functions", of "rapidly decreasing functions" and of C ∞ functions, respectively, see [4,6,15]. S r (R n ) denotes the closed subspace of S (R n ) consisting of radially symmetric temperate distributions. In order to display the active variable in a distribution, say x ∈ R n , we use notation as T (x) or T ∈ D (R n x ). Furthermore, we use at one instance the space s (N 0 ) of slowly increasing sequences, which is the dual of the space s(N 0 ) of fast decreasing sequences.
The spaces D L p , D L p , 1 ≤ p ≤ ∞, were introduced in [15, Ch. VI, Sect. 8, p. 199], and we need, just for p = 1 and p = ∞, so-called "weighted" D L p and D L p -spaces, i.e., D L p ,μ := D L p · (1 + |x| 2 ) −μ/2 , D L p ,μ := D L p · (1 + |x| 2 ) −μ/2 , 1 ≤ p ≤ ∞, μ ∈ R, see [7,8]. (The first appearance of a weighted D L p -space seems to be in [13,Sect. 3,p. 7].) In Sect. 6, we also employ the spaces For the evaluation of a distribution T ∈ E on a test function φ ∈ E (where E is a normal space of distributions, see [6, Ch. IV, Sect. 2, Def. 3, p. 319]), we use angle brackets, i.e., φ, T or, more precisely E φ, T E . More generally, if φ ∈ E⊗ F and T ∈ E for distribution spaces E, F, then E⊗ F φ, T E symbolizes the vector-valued scalar product (E⊗ F) × E → F, see [14] for more information on vector-valued distributions. (In all tensor products of this study, both factors are complete and at least one of the factors is nuclear and hence E⊗ π F = E⊗ F and we simply write The Heaviside function is denoted by Y , see [15, p. 36], and we write δ τ (t) ∈ D (R 1 t ), τ ∈ R, for the delta distribution with support in τ, which is the derivative of In contrast, δ ∈ D (R n ) without any subscript stands for the delta distribution at the origin. For z ∈ C with Re z > −1, we write t z + for the locally integrable function Y (t)t z and we obtain, by analytic continuation, the analytic distribution-valued function We use the Fourier transform F in the form this being extended to S by continuity. For the partial Fourier transforms of a distribution T ∈ S (R m+n xy ) with respect to x or y, respectively, we use the notation F x T and F y T , respectively.

Review of definitions and classical results
The Weyl transformation W can be described as a partial Fourier transform composed with a linear transformation in R 2n xy , to wit see [3, p. 79], [20, Equ.(6.7), p. 30]. Hence if σ ∈ S (R 2n ), then its Weyl transform W σ is the pseudo-differential operator represented by the continuous linear mapping For σ ∈ S(R 2n ) and φ ∈ S(R n ), this implies In this study, we are mainly concerned with the question for which σ the Weyl transform W σ can be extended to a bounded linear mapping W σ : i.e., W σ ∈ L(L 2 (R n )). In other words, we investigate for which σ the norm W σ φ 2 can be bounded by a constant multiple of φ 2 for φ ∈ S(R n ). If this condition is satisfied, we look for criteria guaranteeing that W σ is a compact operator in L 2 (R n ), i.e., that W σ ∈ Com(L 2 (R n )).
Usually, W is expressed by the "Wigner transform," which is a continuous sesquilinear mapping The mappings w and W are connected by

This implies
and hence Note that we use for w a normalization constant different from that in [20,Equ. (3.12), p. 15], and that we also changed, for clarity, the notation for the Wigner transform from W (φ, ψ) to w(φ, ψ). Next note that the properties of w mirror those of W . In fact, the bilinear mapping (φ, ψ) → w(φ,ψ) extends from the tensor product S(R n ) ⊗ S(R n ) to its completion S(R 2n ). This gives rise to the linear mapping which trivially extends to continuous linear mappings from L 2 (R 2n ) into itself, and from S (R 2n ) into itself, respectively. We shall write again w(φ, ψ) forw(φ ⊗ψ) if φ, ψ both belong to L 2 (R n ) or to S (R n ), respectively. Furthermore, from and the density of S(R n ) in L 2 (R n ), we conclude that is well-defined and continuous with respect to the supremum norm on C 0 (R 2n ). This in turn leads to the following proposition. [1,2] [1,2] , it operates as a Hilbert-Schmidt operator on L 2 (R n ). Of course, the inequality implies that W σ can be extended to a continuous linear mapping in L 2 (R n ), i.e., W σ ∈ L(L 2 (R n )). Moreover, Hilbert-Schmidt operators are compact since the functions in L 2 (R n ) ⊗ L 2 (R n ) (which lead to finite rank operators) are dense in L 2 (R 2n ). (b) If σ ∈ M 1 (R 2n ) and φ, ψ ∈ S(R n ), then the inequality in (2.1) implies From the density of S(R 2n ) in L 1 (R 2n ) and since the linear mapping is continuous, we also conclude that W σ is a compact operator for σ ∈ L 1 (R 2n ). Note, however, that L 1 (R 2n ) is closed and thus not dense in and this operator is clearly not compact. (c) This follows from (a) and (b) on account of the inclusion p∈ [1,2] and (c). The proof is complete.
The case 2 < q < ∞ is trickier; it was settled in [16]. The detailed elaboration in [20, Sect. 13, p. 63] consists in an indirect proof relying on the uniform boundedness principle. We shall give a constructive proof in Proposition 2 below by means of an explicit example of σ ∈ q∈ [2,∞] The ndimensional case follows therefrom by tensoring.)

Proposition 2 The function
Proof Evidently, the function (1 + |x|) −1/2 belongs to C 0 (R) ∩ q>2 L q (R). The same holds true for and This shows that W σ φ is not square integrable and thus W σ / ∈ L(L 2 (R)). The proof is complete.

Remark
The known asymptotic of the p-norm (for 1 ≤ p < 2) of the Laguerre functions L j (2t)e −t , see [17, Lemma 1.5.4, p. 28], implies, due to Proposition 3 below and via the uniform boundedness principle, that σ ∈ L q (R 2 ) (for 2 < q ≤ ∞) could even be chosen radially symmetric and such that W σ is unbounded on L 2 (R). We are not aware, however, of an explicit construction for such a function σ.

Rotationally invariant symbols
We assume first that n = 1 and we postpone the straight-forward generalization to higher dimensions n to Sect. 6. We hence consider radially symmetric symbols σ ∈ S (R 2 ) and we aim at establishing criteria for the boundedness or compactness, respectively, of the operators W σ on L 2 (R).
For motivation, let us first describe the two lines of reasoning in [20]: In [20, Sect. 24], σ (x, y) is a temperate radial function and it is shown that the Hermite functions h j ∈ S(R) (for the notation see Sect. 1) are eigenfunctions of W σ , i.e., W σ h j = λ j h j , λ j ∈ C, j ∈ N 0 . Therefore the mapping properties of W σ are characterized by the eigenvalues λ j and it is shown that where L j denotes Laguerre polynomials of order 0, see [20,Thm. 24 [20,Thm. 26.1,p. 123]. We shall take up this second line of reasoning in Sect. 4 below, and we shall generalize the lastmentioned theorem in [20, p. 123] in order to give a characterization of boundedness and compactness, respectively, of W σ . Thereupon, we shall analyze some illustrative examples in Sect. 5.
In this section, we slightly generalize Theorem 24.5 in [20, p. 115] (and in particular formula (3.1) for the eigenvalues λ j ) from temperate radial functions to temperate radial distributions.

Proposition 3
Let h j and L j , j ∈ N 0 , denote the Hermite functions and Laguerre polynomials, respectively, see Sect. 1, and let be as in (3.4).
Proof For a temperate function S, we have σ = (S) = S(x 2 + y 2 ) and formula For the remaining assertions in Proposition 3, we follow the reasoning in [20,Sect. 24]. We note that e j := h j −1 2 h j , j ∈ N 0 , yields an orthonormal basis in L 2 (R) such that ∞ j=0 φ, e j e j converges not just in L 2 (R) but also in S(R) to φ for φ ∈ S(R), compare [ and thus W σ φ 2 ≤ φ 2 · sup j∈N 0 |λ j |, W σ ∈ L(L 2 (R)) and W σ L(L 2 (R)) = sup j∈N 0 |λ j |. Finally, it is well-known that a bounded diagonal operator is compact if and only if the series of its eigenvalues converges to 0. This completes the proof.

Symbols represented by radial Fourier transformation
Since the mapping h(x, y) = x 2 + y 2 (compare (3.2)) is not submersive at 0, some care has to be exercised when composing a distribution S ∈ D (R) with h. In Sect. 3, we took recourse to the space S ([0, ∞)), here we shall consider S = F T for T ∈ D L 1 ,−1 (R). Then we can define σ = h * S = (F T )(x 2 + y 2 ) ∈ S (R 2 xy ) by the formula D L ∞ ,1⊗ S (R 2 ) , see the notation in Sect. 1. Of course, in R 2 \{0}, the distribution σ coincides with the pullback h * S defined in (3.3).
Let h j denote the Hermite functions as in Sect. 1. Then W σ h j = λ j h j where In particular, W σ ∈ L(L 2 (R)) if and only if the sequence (λ j ) j∈N 0 is bounded, and W σ ∈ Com(L 2 (R)) if and only if lim j→0 λ j = 0.
Proof Under somewhat stronger assumptions on σ, formula (4.2) for the eigenvalues λ j was already deduced in [20,Sects. 25, 26] from (3.5) by Fourier transformation. We prefer to give a direct, independent proof of (4.2), which in turn would then also yield (3.5) by the residue theorem.
If t = 0 and σ t (x, y) = e −it(x 2 +y 2 ) , then is an isomorphism and since e −it(x 2 +y 2 ) ∈ D L ∞ ,1 (R)⊗ S (R 2 ) by [10, Lemma 2.2, p. 266], we conclude that the kernel K (t, x, y) = W (e −it(x 2 +y 2 ) ) also belongs to D L ∞ ,1 (R)⊗ S (R 2 ). For t = 0, it is given by the C ∞ function in (4.3). (Of course, K also depends C ∞ on t ∈ R with values in S (R 2 ) and K (0, x, y) = W 1 (x, y) = δ(y − x).) This implies that When applying W σ to a test function φ ∈ S(R) we therefore conclude by means of Fubini's Theorem, see [14, Cor., p. 136], that We shall show below that and this will imply then formula (4.2). The remaining statements in Proposition 4 follow therefrom as in Proposition 3. For t = 0, formula (4.4) is immediate, and for t = 0, we infer from (4.3) the following: . This concludes the proof.

Proof (a) In fact, if
, then the eigenvalues λ j of W σ are bounded by μ M 1 due to formula (4.2): and therefore W σ ∈ Com(L 2 (R)) for T ∈ L p (R) due to (b). The proof is complete.

Remark
The last corollary generalizes Theorem 26.1 in [20, p. 123]. In fact, there the compactness of W σ is proven for T ∈ ∪ p∈ [1,2] L p (R) and for T ∈ ∪ p∈ [1,2] F(L p (R)). Note that the first assertion is contained in (c) of the corollary. The second one follows directly from Proposition 1 (c) since

Three examples
In the following three examples, the radially symmetric distributions σ z (x, y) ∈ S (R 2 xy ) will depend meromorphically on the complex parameter z, see [8] for the theoretical background. Since the mapping W is linear and continuous, we infer that the eigenvalues λ j (z) are also meromorphic functions of z and can be determined by analytic continuation.
Example 1 Let us start with σ z (x, y) = (x 2 + y 2 ) z . This yields an analytic function with simple poles at z ∈ −N and residues and hence σ z (x, y) = (F T z )(x 2 + y 2 ) holds in the sense of formula (4.1). Therefore, formula (4.2) in Proposition 4 furnishes, if z ∈ C\N 0 with Re z > −1, the representation for the eigenvalues of W σ z .
Example 2 Let us fix ρ > 0. Since (t − ρ 2 ) z + ∈ S (R) depends analytically on z ∈ C\(−N), see [9, Ex. 1.4.8, p. 49], and since the mapping h(x, y) = x 2 + y 2 is submersive at the circle h = ρ 2 , we obtain an analytic function At z = −k, k ∈ N, the function σ has simple poles with the residues In order to study the asymptotic behavior of the eigenvalues λ j (z), j ∈ N 0 , of W σ z , we first assume that Re z > −1.
see [9, Ex. 1.6.7, p. 86], and hence formula (4.2) in Proposition 4 yields For −1 < Re z < 0, formula (5.3) represents λ j (z) by an absolutely convergent integral. If we use and substitute t = cot x, then we obtain Formula (5.4) remains valid for C\(−N) if the distribution S in square brackets is defined by analytic continuation with respect to z. Therefore the asymptotic behavior of λ j (z) is determined by the behavior of the distribution S at its singular support, i.e., at the three points ± π 2 and 0. The points ± π 2 yield, similarly as in Example 1, the asymptotic behavior (2 j) z . Let us consider now the origin. For j → ∞, it contributes the same asymptotic behavior as In particular, this implies that W σ z ∈ Com(L 2 (R)) if and only if − 3 2 < Re z < 0 and W σ z ∈ L(L 2 (R)) if and only if − 3 2 ≤ Re z ≤ 0, z = −1. As a control, we can consider the limit and obtain the correct asymptotic behavior, compare the example in Sect. 3.

Example 3 Let us finally consider the symbols
if χ ∈ C ∞ (R) with χ(t) = 1 for t ≥ 2, χ(t) = 0 for t ≤ 1. (Since (x 2 + y 2 ) z has already been investigated in Example 1, we are interested here only in the contributions stemming from the behavior of σ p,z near infinity and we therefore employ the excision function χ.) we can apply formula (4.2) in Proposition 4. The distribution T is again a fast oscillating C ∞ function whose behavior at infinity is readily obtained by the method of stationary phase: If t < 0 and h(s) = s p + st, 1 p + 1 q = 1 and h (s 0 ) = 0, then For t → ∞, the function T (t) is fast decreasing.
With the same substitution as in formula (5.4), we therefore obtain for the eigenvalues λ j ( p, z) of W σ p,z the following: (For the convergence of the last integral, we have to stipulate that Re z < 3 2 p − 1. The final result in formula (5.5) below can, as before, be extended to z ∈ C by analytic continuation. This is done by representing the integral as the Fourier transform of a distribution S and by employing the equation Upon using again the method of stationary phase with h(x) = 2 j x +( p−1)( px) −q and px 0 = (2 j/ p 2 ) (1− p)/ (2 p−1) , this finally implies

Generalization to n dimensions
Let us consider here the variables in the ordering x 1 , y 1 , x 2 , y 2 , . . . , x n , y n . We denote by S pr (R 2n ) the set of all partially radially symmetric σ ∈ S (R 2n ), i.e., those temperate distributions σ that are rotationally symmetric with respect to all (x j , y j )−planes, i.e., such that for all two-dimensional rotations A 1 , . . . , A n . As in Sect. 3, we obtain an isomorphism Alternatively, we can consider σ = (F T )(x 2 1 + y 2 1 , . . . , x 2 n + y 2 n ) for As in formula (4.1), σ is defined by for φ ∈ S(R 2n ). Then Propositions 3 and 4 generalize to the following.

Proposition 5
Let h α , α ∈ N n 0 , and L j , j ∈ N 0 , denote the Hermite functions and Laguerre polynomials, respectively, see Sect. 1, and let pr be as in (6.1).
for t ∈ R n , then Proposition 4 yields and this implies formula (6.4) as in the proof of Proposition 4. Formula (6.3) can be deduced from formula (6.4) , approximation yields that formula (6.4) can be applied in the form where the integrals are conditionally convergent. Eventually, formula (6.3) follows upon employing the residue theorem, see the details in the proof of the corollary below. Alternatively, formula (6.3) follows directly from Proposition 3 for tensor products S ∈ S ([0, ∞)) ⊗ · · · ⊗ S ([0, ∞)) and then must hold generally by density. This completes the proof.
Let us finally specialize to symbols σ in the space S r (R 2n ) = {σ ∈ S (R 2n ); σ is radially symmetric with respect to all variables}.
According to [12] or [18, Sect. 3.10, p. 249], the isomorphism can be extended to R 2n in the following way: Corollary Let h α , α ∈ N n 0 , and L n−1 j , j ∈ N 0 , denote the Hermite functions and Laguerre polynomials, respectively, see Sect. 1, and let r be as in (6.5).