On the boundedness of the Weyl transform for homogeneous distributions on R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^2$$\end{document}

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Introduction and notation
In our first paper on Weyl transforms, see [8], we endeavored to generalize the theorems and examples in [11] concerning the boundedness and compactness, respectively, of Weyl transforms of radially symmetric functions to the case of radially symmetric distributions. (Incidentally, let me indicate three printing errors in [8]: On p. 769, the space D L 1 (R 2 ) should be replaced by D L 1 (R 1 ) and on p. 775, the expression e i exp(2x) was twice contorted to e i exp(2x); finally, in line 13 on p. 785, the factor s z 0 got lost in the expression (2π) −1 s z 0 e ih(s 0 ) 2π i/h (s 0 ).) In [8], we did not touch on Theorem 27.3 on tensor products in polar coordinates in [11, p. 128] for a variety of reasons: First, this theorem concerns a much more general situation (of which radially symmetric functions are just a tiny special case), second, the setting is much more intricate and arduous than for radially symmetric functions, third, we did not immediately succeed in finding a generalization to distributions with meaningful and manageable examples, and fourth, there is no hope (as far as we can see) for a single necessary and sufficient criterion for boundedness, in contrast to the radial case.
This paper tries to work out [11,Thm. 27.3,p. 128] from a distribution-theoretic viewpoint. We give a distributional formulation of it in Proposition 1, and we specialize Proposition 1 to the case of homogeneous distributions σ on R 2 in Proposition 4. Therein we state a sufficient condition for the boundedness of the Weyl transform W σ on L 2 (R) in terms of the Fourier coefficients of the characteristic F of σ. In the preparation for the general case, we deal with the case of homogeneity 0 in Proposition 2, where the matrix coefficients of the Weyl transforms can be expressed more explicitly than in the general case. In Proposition 3, we give necessary conditions on the degree λ of homogeneity and on the characteristic F of σ = F · r λ that must be satisfied in order that W σ can be bounded. Let us introduce some notation. Besides the familiar Banach spaces L p (R n ), 1 ≤ p ≤ ∞, we use the Banach spaces 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. 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 [4,9]. 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 ). For the evaluation of a distribution T ∈ E on a test function φ ∈ E, we use angle brackets, i.e., φ, T or, more precisely E φ, T E .
N, N 0 and Z denote the sets of positive, of non-negative, and of all integers, respectively. For sequences with the index sets N 0 and Z, respectively, we use the Banach spaces l p , 1 ≤ p ≤ ∞, and the Fréchet space s of fast decreasing sequences with its dual s of slowly increasing sequences.
S 1 denotes the unit circle in R 2 , which we identify with C, and we write ω for the generic variable on S 1 ⊂ C and dω for the measure of length on S 1 . With these notations, The Heaviside function is denoted by Y , see [9, 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 Y (t − τ ), i.e., φ, δ τ = φ(τ ) for φ ∈ D(R 1 ). 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 § by continuity. For the partial Fourier transforms of a distribution T ∈ S (R m+n xy ) with respect to x ∈ R m or y ∈ R n , respectively, we use the notation F x T and F y T , respectively.
The Hermite polynomials H j and Hermite functions h j are defined as usually: see [5,Section 8.95], [11,Sections 18,19]. Similarly, the Laguerre polynomials L α j (x) of order α are defined by

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 The Weyl transform W can be expressed by the "Wigner transform," which is a continuous sesquilinear mapping The mappings w and W are connected by Note that we use for w a normalization constant different from that in [11,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).
In this study, we shall focus on symbols σ which are homogeneous distributions in two dimensions, and we shall derive necessary and sufficient criteria, respectively, for the boundedness of the corresponding pseudo-differential W σ on L 2 (R 1 ), i.e., criteria on whether W σ can be extended to a bounded mapping L 2 (R) → L 2 (R), or, in other words, whether the norm W σ φ 2 can be bounded by a constant multiple of φ 2 for φ ∈ S(R 1 ).
Let us now recapitulate some basic facts about homogeneous distributions σ on R 2 , see [6,7] for more details. If σ ∈ D (R 2 \{0}) is homogeneous of degree λ ∈ C, then the structure theorem (see [6,Sect. 2.5]) implies that σ can be extended to R 2 and can be represented in the form σ = F · r λ where the so-called "characteristic" F belongs to D (S 1 ) and If we identify S 1 ⊂ R 2 C, then 2π ω −k , F ∈ C for k ∈ Z and k∈Z c k ω k converges to F(ω) in D (S 1 ). The Hilbert spaces are the usual Sobolev spaces on S 1 .
The distribution-valued function z → t z + is meromorphic with simple poles at z = −k, k ∈ N. We have Res z=−k t z We finally observe that the factor in front of c k in formula (2.2) does not depend on the sign of k on account of the complement formula for the gamma function.
As we shall see in Proposition 3 below, a homogeneous symbol σ ∈ S (R 2 ) can have a bounded Weyl transform W σ only under the condition that the degree λ of homogeneity fulfills −2 ≤ Re λ ≤ 0. For homogeneous and radially symmetric symbols σ = r λ ∈ S (R 2 ), we have shown in [8,Sect. 5 Let us describe next what can be inferred from classical results. According to [11, Thms. 11.1, 11.3, pp. 55, 57], compare also [8, Prop. 1, p. 773], W σ is bounded and even compact as an operator on L 2 (R n ) if σ or Fσ belong to ∪ p∈ [1,2] , then we employ formula (2.2) and conclude once again that W σ ∈ Com(L 2 (R 1 )) from the asymptotic behaviour . We shall deduce more general results in Sect. 6 below.

The matrix representation of W
The normalized Hermite functions are an orthonormal basis in L 2 (R), see, e.g., [11,Thm. 19.3,p. 94]. This yields an isomorphism of Hilbert spaces: As was observed already by Schwartz, cf. [9, Ch. VII, Sect. 7, Ex. 7, pp. 260, 262], the sequence e j , j ∈ N 0 , also serves to identify the spaces S(R) and S (R) with the sequence spaces s(N 0 ) and s (N 0 ), respectively. Hence we obtain a commutative diagram: Here the vertical isomorphisms in the diagram are all given by T → ( e j , T ) j∈N 0 .
By the same token, using S ( on account of formula (2.1). Therefore, the operator W σ is bounded on L 2 (R) if and only if holds.
Let us specialize now to distributions σ ∈ S (R 2 ) that are tensor products in polar coordinates. For the radial dependency of σ, we take R ∈ S ([0, ∞)), which is the As in [8, p. 776], we observe that is an isomorphism of locally convex spaces.
If, furthermore, F ∈ D (S 1 ), then we define F · R ∈ S (R 2 ) by the following: Note that this is well-defined since the function is not uniquely determined up to a factor by the distribution σ = F · R since, e.g., F · δ = 0. In the following Proposition 1, we state formulas for the matrix coefficients of W σ with respect to the normalized Hermite functions e j , e k if σ = F · R ∈ S (R 2 ) is a tensor product in polar coordinates. This proposition is but a slight extension of Theorem 27.3 in [11, p. 128] from temperate functions to temperate distributions. Proposition 1 Let e j be the normalized Hermite functions as in formula (3.1) and L α k be the Laguerre polynomials as in Sect.
being the matrix representation of W σ and define the symmetric matrix (a jk ) j,k∈N 0 ∈ s (N 2 0 ) by
In the second part, we will closely follow the proof of Theorem 27.3 in [11, p. 129]. Assuming condition (3.5) and (1+|k|) μ c k k∈Z ∈ l 1 (Z), we set d k = C(1+|k|) μ |c k |, k ∈ Z. Then the sequence (d k ) k∈Z belongs to l 1 (Z) and i.e., B σ and thus also W σ are bounded. This completes the proof.
If we set F = 1 in Proposition 1, we obtain c k = δ k0 and hence the matrix B σ is diagonal with the eigenvalues 2), p. 779], a representation of the eigenvalues λ j = a j j , j ∈ N 0 , by integrals with rational kernels was derived for σ ∈ S (R 2 ) radially symmetric and such that σ (x, y) = (F T )(x 2 + y 2 ) with T ∈ D L 1 ,−1 (R). This formula, i.e., was useful in some examples in calculating the asymptotic behavior of λ j for j → ∞, see [8,Sect. 5].
Since we will make no use here of the corresponding "rational" representation of a jk , let us just state the formula without proof: If j, k ∈ N 0 , j ≥ k, T ∈ D L 1 ,−1 (R) and R(r ) = (F T )(r 2 ) · r j−k holds in S ([0, ∞)), then (3.9) As in the diagonal case, formula (3.9) can be derived from formula (3.4) by Fourier transformation, compare the procedure in [11,Sects. 25,26].

The case of homogeneous symbols of degree 0
We specialize now our investigation to homogeneous symbols σ = F · r λ , λ ∈ C, F(ω) = k∈Z c k ω k ∈ D (S 1 ). We assume that σ is homogeneous on R 2 and hence that c k = 0 for |k| ≤ j if λ = −2 − j, j ∈ N 0 , cf. Sect. 2.
In order to calculate a jk (λ) for R(r ) = r λ + ∈ S ([0, ∞)) according to formula (3.4), we first assume that λ ∈ C\Z with Re λ > −2. Then [1, Equ. 11.44, p. 110] and [5, Equ. 9.131.1] yield the following for j, k ∈ N 0 with j ≥ k : By analytic continuation, formulas (4.1), (4.2) hold for λ ∈ C\Z and, due to the symmetry of (4.2) in j, k, for all j, k ∈ N 0 . But note that the hypergeometric function Because of that and because of the gamma factor in (4.1), (4.2), we have to exercise some care if λ ∈ Z.
Let us now specialize further and consider the case λ = 0, i.e., the case of homogeneous symbols σ = F(ω) of degree 0.
Proof (a) If j = k ∈ N 0 , then formula (4.1) yields a j j (0) = 1. If j = k, then formula (4.2) is valid. Let us first assume j > k and k even. For Re z < −k, [2, Equ. 7.3.7.4] yields and hence on account of the complement formula and of the duplication formula for the gamma function.
(b) Let us next estimate a jk (0). Using Euler's psi function ψ(x) = (log (x)) for x > 0 we obtain Equation (4.4) implies, incidentally, that α j is monotonically increasing since the same holds for the function ψ due to the logarithmic convexity of the gamma function, see, e.g., [5,Equ. 8.363.3].
Since C 1 + log x ≤ ψ(x) ≤ C 2 + log x holds for all x ≥ 1 2 and appropriate constants C 1 , C 2 ∈ R, we conclude that This implies Therefore, condition (3.5) in Proposition 1 holds for μ = 1 4 and this implies that W σ is bounded on L 2 (R) if the condition ( 4 √ |k| c k ) k∈Z ∈ l 1 (Z) is assumed. The proof is complete.

Necessary conditions
We shall prove here that σ ∈ S (R 2 ) homogeneous of degree λ can have a bounded Weyl transform only if the real part of λ lies in the interval [−2, 0] and if the characteristic F of σ belongs to the Sobolev space H (1/4+μ/2) (S 1 ). In deriving this result, we need yet another representation of a jk (λ) by the hypergeometric function.
If we use the second equation in [1,Equ. 11.44,p. 110] or, alternatively, apply (employing a limit procedure) [5,Equ. 9.131.2] to formula (4.1), then we obtain the following for λ ∈ C\Z and j, k ∈ N 0 with j ≥ k : Let us now digress a little bit and derive the "complement formula" for the matrix coefficients a jk (λ) from formula (5.1). Upon applying [5,Equ. 9.131.1] to formula (5.1) we obtain, still for j ≥ k, wherefrom we conclude that Note that the factor in (5.2) in front of a jk (−2 − λ) is symmetric in j and k. We observe that the special case of j = k in (5.2) was already given in [8, p. 782] and that formula (5.2) implies a jk (−1) = 0 if the minimum of j and k is odd, compare the remark following Proposition 2. Also note that formula (5.2) is a consequence of formula (2.2) and of the fact that W σ and W F σ are closely connected.
Proof (a) Let us first settle the case of homogeneous σ with support contained in {0}. Then λ = −2 − j, j ∈ N 0 , and σ is a linear combination of ∂ k x ∂ l y δ for k, l ∈ N 0 with k + l = j. This implies that W σ is a linear combination of the operators A kl , and the boundedness of A kl implies k = 0, and and the boundedness of A 0l implies l = 0 and hence σ is a multiple of δ.
(b) Let now σ = F · r λ with F(ω) = k∈Z c k ω k ∈ D (S 1 ) and assume that c m = 0 for some m ∈ Z. In particular, in the case of λ = −2 − j, j ∈ N 0 , the homogeneity of σ on R 2 implies that |m| > j.
Since W σ ∈ L(L 2 (R)), its matrix coefficients b jk = e j , W σ e k are bounded in modulus by W σ L(L 2 (R)) . Hence also the complex sequence a k,k+m (λ) = b k,k+m /c m , k ∈ N 0 , k ≥ −m, must be bounded.
Let us next determine the asymptotic behavior of a k+m,k (λ) for k → ∞ and m ∈ N 0 fixed, and for λ ∈ C\(−2 − N 0 ) or such that λ = −2 − j, j ∈ N 0 , and m > j.
Upon inserting the standard integral representation for the hypergeometric function (see [5,Equ. 9.111]) into formula (5.1) and using analytic continuation, we obtain Note that t is an entire distribution-valued function of λ which can be multiplied with (1−t) (m+λ)/2 + since the singular supports of the two distributions are disjoint. Also note that (1 − t) (m+λ)/2 + is analytic at the value of λ which we consider. We then obtain the asymptotic behavior of a k+m,k (λ) for k → ∞ by approximating (1 − 2t) k in formula (5.3) by (1 − t) 2k near zero and by (−t 2 ) k near one, respectively. This implies Hence, eventually, We remark that the special case of m = 0 was shown already in [8, p. 783].
. Altogether we conclude that F belongs to the Sobolev space H (1/4+μ/2) (S 1 ) and this completes the proof.

Remark
Let us yet comment on the situation for symbols σ that are homogeneous on R 2 \{0}, but only associated homogeneous on R 2 , i.e., σ = F · r −2− j , j ∈ N 0 , and F(ω) = k∈Z c k ω k with c k = 0 for some k ∈ Z with |k| ≤ j. In this case, W σ cannot be bounded on L 2 (R). We have omitted this case in Proposition 3 since the asymptotic expansion in formula (5.4) becomes slightly more complicated. As to be expected, logarithms appear. Just to get an idea, let us analyze the case of homogeneity λ = −2.
According to the complement formula (5.2), we obtain In particular, this implies On the other hand, a j j (−2) grows logarithmically for j → ∞ and thus W σ is not bounded if σ = r −2 = Pf λ=−2 r λ . In fact, setting m = 0 and k = j ∈ N in formula (5.3) yields .

A sufficient condition ensuring the boundedness of W
When comparing the three examples λ = 0 (in Proposition 2), λ = −1 (in the remark following Proposition 2) and λ = −2 (in the remark following Proposition 3), we see that we have to stipulate an l 1 condition on the Fourier coefficients c k of the characteristic F of the symbol σ = F · r λ in order to guarantee that W σ is bounded.
The following proposition incorporates these special cases into a general statement.
At the end of Sect. 2, we have seen that in the case of −2 < μ = Re λ < −1, assumption (A) implies the boundedness of W σ on L 2 (R) for σ = F · r λ . Instead, assumption (B) implies that W σ is bounded whenever μ ∈ [−2, 0] according to Proposition 4. If −2 < μ < − 3 2 , then (1 + |k|) 1/4+μ/2 k∈Z ∈ l 2 (Z) and hence (A) implies (B). So condition (B) is the less restrictive and more general one. In contrast, when (1 + 2 j ) −1/2 2 j/2 j −2 ≤ ∞ j=1 j −2 < ∞ but (c k ) k∈Z / ∈ l 2 (Z). Hence (B) holds and (A) fails. As was pointed out already in the introduction, in contrast to the case of radially symmetric distributions, we cannot hope for a condition that is both necessary and sufficient for the boundedness of W σ if σ is a tensor product in polar coordinates. This is a consequence of the fact that the Hermite functions in this case do not yield eigenfunctions of W σ . So the approach in Proposition 1 has its limitations from the word go.
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