On the boundedness of the Weyl transform for homogeneous distributions on ${\mathbb {R}}^2$

Abstract

Starting from earlier work by M.W. Wong, we give necessary and sufficient criteria, respectively, for the boundedness of the Weyl transform $W_\sigma$ as an operator on $L^2({\mathbb {R}})$ if $\sigma$ is a homogeneous distribution on ${\mathbb {R}}^2$.

1 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 $\mathcal D'_{L^1}({{\mathbb {R}}}^2)$ should be replaced by $\mathcal D'_{L^1}({{\mathbb {R}}}^1)$ and on p. 775, the expression $\text {e}^{\text {i}\,\exp (2x)}$ was twice contorted to $\text {e}^{\text {i}}\exp (2x)$; finally, in line 13 on p. 785, the factor $s_0^z$ got lost in the expression $(2\pi )^{-1}s_0^z\,\text {e}^{\text {i} h(s_0)}\sqrt{2\pi \text {i}/h''(s_0)}.)$

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 $\sigma$ on ${{\mathbb {R}}}^2$ in Proposition 4. Therein we state a sufficient condition for the boundedness of the Weyl transform $W_\sigma$ on $L^2({{\mathbb {R}}})$ in terms of the Fourier coefficients of the characteristic F of $\sigma .$ 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 $\lambda$ of homogeneity and on the characteristic F of $\sigma =F\cdot r^\lambda$ that must be satisfied in order that $W_\sigma$ can be bounded.

Let us introduce some notation. Besides the familiar Banach spaces $L^p({{\mathbb {R}}}^n),$$1\le p\le \infty ,$ we use the Banach spaces of bounded and of compact linear operators on $L^2({{\mathbb {R}}}^n),$ respectively, which we denote by $\mathcal L(L^2({{\mathbb {R}}}^n))$ and by $\text {Com}(L^2({{\mathbb {R}}}^n)),$ respectively.

We employ the standard notation for the distribution spaces $\mathcal D',\,\mathcal S',\,\mathcal E',$ the dual spaces of the spaces $\mathcal D,\,\mathcal S,\,\mathcal E$ of “test functions”, of “rapidly decreasing functions” and of $C^\infty$ functions, respectively, see [4, 9]. In order to display the active variable in a distribution, say $x\in {{\mathbb {R}}}^n,$ we use notation as T(x) or $T\in \mathcal D'({{\mathbb {R}}}^n_x).$ For the evaluation of a distribution $T\in E'$ on a test function $\phi \in E,$ we use angle brackets, i.e., $\langle \phi ,T\rangle$ or, more precisely ${}_E\langle \phi , T\rangle _{E'}.$

The spaces $\mathcal D_{L^p},\mathcal D'_{L^p},\ 1\le p\le \infty ,$ were introduced in [9, Ch. VI, § 8, p. 199], and we need, just for $p=1$ and $p=\infty ,$ so-called “weighted” $\mathcal D_{L^p}\!$ and $\mathcal D'_{L^p}\!$-spaces, i.e.,

$$\begin{aligned} \mathcal D_{L^p,\mu }:= & {} \mathcal D_{L^p}\cdot (1+|x|^2)^{-\mu /2},\ \mathcal D'_{L^p,\mu }:=\mathcal D'_{L^p}\cdot (1+|x|^2)^{-\mu /2},\\&1\le p\le \infty ,\ \mu \in {{\mathbb {R}}}, \end{aligned}$$

see [6].

${{\mathbb {N}}},$${{\mathbb {N}}}_0$ and ${{\mathbb {Z}}}$ denote the sets of positive, of non-negative, and of all integers, respectively. For sequences with the index sets ${{\mathbb {N}}}_0$ and ${{\mathbb {Z}}},$ respectively, we use the Banach spaces $l^p,$$1\le p\le \infty ,$ and the Fréchet space s of fast decreasing sequences with its dual $s'$ of slowly increasing sequences.

${{\mathbb {S}}}^1$ denotes the unit circle in ${{\mathbb {R}}}^2,$ which we identify with ${{\mathbb {C}}},$ and we write $\omega$ for the generic variable on ${{\mathbb {S}}}^1\subset {{\mathbb {C}}}$ and $\text {d}\omega$ for the measure of length on ${{\mathbb {S}}}^1.$ With these notations, $(2\pi )^{-1/2}\omega ^k,$$k\in {{\mathbb {Z}}},$ is an orthonormal basis in $L^2({{\mathbb {S}}}^1)$ and $f(\omega )= \sum _{k\in {{\mathbb {Z}}}}c_k\omega ^k$ is the usual Fourier series expansion for $f\in \ L^2({{\mathbb {S}}}^1)$ if $c_k=\frac{1}{2\pi }\int _{{{\mathbb {S}}}^1} f(\omega )\,\omega ^{-k}\,\text {d}\omega ,$$k\in {{\mathbb {Z}}}.$ Furthermore, $L^1({{\mathbb {S}}}^1)$ is embedded into $\mathcal D'({{\mathbb {S}}}^1)$ by $\langle \phi ,f\rangle =\int \phi (\omega )f(\omega )\,\text {d} \omega .$

The Heaviside function is denoted by Y,  see [9, p. 36], and we write $\delta _\tau (t)\in \mathcal D'({{\mathbb {R}}}^1_t),\ \tau \in {{\mathbb {R}}},$ for the delta distribution with support in $\tau ,$ which is the derivative of $Y(t-\tau ),$ i.e., $\langle \phi ,\delta _\tau \rangle =\phi (\tau )$ for $\phi \in \mathcal D({{\mathbb {R}}}^1).$ In contrast, $\delta \in \mathcal D'({{\mathbb {R}}}^n)$ without any subscript stands for the delta distribution at the origin. For $z\in {{\mathbb {C}}}$ with $\text {Re}\,z>-1,$ we write $t_+^z$ for the locally integrable function $Y(t)t^z$ and we obtain, by analytic continuation, the analytic distribution-valued function

$$\begin{aligned} t_+^z:{{\mathbb {C}}}{\setminus }(-{{\mathbb {N}}})\longrightarrow \mathcal S'({{\mathbb {R}}}^1_t):z\longmapsto t_+^z, \end{aligned}$$

see [7, Ex. 1.4.8, p. 49].

The pull-back $h^*T=T\circ h\in \mathcal D'(\Omega )$ of a distribution T in one variable t with respect to a submersive $\mathcal C^\infty$ function $h:\Omega \rightarrow {{\mathbb {R}}},\ \Omega \subset {{\mathbb {R}}}^n\text { open},$ is defined as in [4, Equ. (7.2.4/5), p. 82] or in [7, Def. 1.2.12, p. 19], i.e.,

$$\begin{aligned} \langle \phi ,h^*T\rangle =\Big \langle \frac{\text {d}}{\text {d}t}\biggl (\int _\Omega Y(t-h(x)) \phi (x)\,\text {d}x\biggr ), T\Big \rangle ,\quad \phi \in \mathcal D(\Omega ). \end{aligned}$$

We use the Fourier transform $\mathcal F$ in the form

$$\begin{aligned} (\mathcal F\phi )(\xi ):= \int _{{{\mathbb {R}}}^n}\text {e}^{-\text {i}\xi x}\phi (x)\,\text {d} x,\quad \phi \in \S ({{\mathbb {R}}}^n), \end{aligned}$$

this being extended to $\S '$ by continuity. For the partial Fourier transforms of a distribution $T\in \mathcal S'({{\mathbb {R}}}^{m+n}_{xy})$ with respect to $x\in {{\mathbb {R}}}^m$ or $y\in {{\mathbb {R}}}^n,$ respectively, we use the notation $\mathcal F_xT$ and $\mathcal F_yT,$ respectively.

The Hermite polynomials $H_j$ and Hermite functions $h_j$ are defined as usually:

$$\begin{aligned} H_j(x)=(-1)^j\text {e}^{x^2}\cdot (\text {e}^{-x^2})^{(j)},\quad h_j(x)=\text {e}^{-x^2/2}H_j(x),\quad j\in {{\mathbb {N}}}_0,\ x\in {{\mathbb {R}}}, \end{aligned}$$

see [5, Section 8.95], [11, Sections 18, 19]. Similarly, the Laguerre polynomials $L^\alpha _j(x)$ of order $\alpha$ are defined by

$$\begin{aligned} L^\alpha _j(x):=j!^{-1} x^{-\alpha }\text {e}^x\cdot (\text {e}^{-x}x^{\alpha +j})^{(j)},\quad x>0,\ \alpha >-1,\ j\in {{\mathbb {N}}}_0, \end{aligned}$$

see [5, Section 8.97], [11, Section 20, p. 95].

2 Review of definitions and classical results

The Weyl transformation W can be described as a partial Fourier transform composed with a linear transformation in ${{\mathbb {R}}}^{2n}_{xy},$ to wit

$$\begin{aligned} W:\mathcal S'({{\mathbb {R}}}^{2n})\longrightarrow \mathcal S'({{\mathbb {R}}}^{2n}):\sigma (x,y)\longmapsto W_\sigma (x,y). \end{aligned}$$

where

$$\begin{aligned} W_\sigma (x,y)=(2\pi )^{-n}(\mathcal F_y \sigma )\Bigl (\frac{x+y}{2},y-x\Bigr ), \end{aligned}$$

see [3, p. 79], [11, Equ. (4.2), p. 19], [8, Sect. 2]. Hence if $\sigma \in \mathcal S'({{\mathbb {R}}}^{2n}),$ then its Weyl transform $W_\sigma$ is the pseudo-differential operator represented by the continuous linear mapping $W_\sigma :\mathcal S({{\mathbb {R}}}^n)\rightarrow \mathcal S'({{\mathbb {R}}}^n)$ with the kernel $W_\sigma (x,y)\in \mathcal S'({{\mathbb {R}}}^{2n}),$ i.e.,

$$\begin{aligned} (W_\sigma \phi )(x)=\langle \phi (y),W_\sigma (x,y)\rangle \in \mathcal S'({{\mathbb {R}}}^n_x),\quad \phi \in \mathcal S({{\mathbb {R}}}^n). \end{aligned}$$

The Weyl transform W can be expressed by the “Wigner transform,” which is a continuous sesquilinear mapping

$$\begin{aligned} w:\mathcal S({{\mathbb {R}}}^n)\times \mathcal S({{\mathbb {R}}}^n)\longrightarrow \mathcal S({{\mathbb {R}}}^{2n}). \end{aligned}$$

The mappings w and W are connected by

$$\begin{aligned} \langle {{\bar{\psi }}},W_\sigma \phi \rangle =\langle w( \phi ,\psi ),\sigma \rangle ,\quad \phi ,\psi \in \mathcal S({{\mathbb {R}}}^n). \end{aligned}$$

(2.1)

This implies

$$\begin{aligned} \begin{aligned} \langle {{\bar{\psi }}},W_\sigma \phi \rangle&=(2\pi )^{-n} \big \langle \overline{\psi (x)}\phi (y), (\mathcal F_y \sigma )\bigl (\tfrac{x+y}{2},y-x\bigr )\big \rangle \\&=(2\pi )^{-n} \big \langle \overline{\psi \bigl (x-\tfrac{y}{2}\bigr )}\phi \bigl (x+\tfrac{y}{2}\bigr ),\mathcal F_y \sigma \big \rangle \\&=(2\pi )^{-n} \Big \langle \mathcal F_y\Bigl ( \overline{\psi \bigl (x-\tfrac{y}{2}\bigr )}\phi \bigl (x+\tfrac{y}{2}\bigr )\Bigr ), \sigma \Big \rangle =\langle w( \phi ,\psi ),\sigma \rangle \end{aligned} \end{aligned}$$

and hence

$$\begin{aligned} \bigl (w(\phi ,\psi )\bigr )(x,y)=(2\pi )^{-n}\int _{{{\mathbb {R}}}^n}\text {e}^{-\text {i}y\eta }\phi (x+\tfrac{\eta }{2})\overline{\psi (x-\tfrac{\eta }{2})}\,\text {d}\eta . \end{aligned}$$

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 $W(\phi ,\psi )$ to $w(\phi ,\psi ).$

In this study, we shall focus on symbols $\sigma$ which are homogeneous distributions in two dimensions, and we shall derive necessary and sufficient criteria, respectively, for the boundedness of the corresponding pseudo-differential $W_\sigma$ on $L^2({{\mathbb {R}}}^1),$ i.e., criteria on whether $W_\sigma$ can be extended to a bounded mapping $L^2({{\mathbb {R}}})\rightarrow L^2({{\mathbb {R}}}),$ or, in other words, whether the norm $\Vert W_\sigma \phi \Vert _2$ can be bounded by a constant multiple of $\Vert \phi \Vert _2$ for $\phi \in \mathcal S({{\mathbb {R}}}^1).$

Let us now recapitulate some basic facts about homogeneous distributions $\sigma$ on ${{\mathbb {R}}}^2,$ see [6, 7] for more details. If $\sigma \in \mathcal D'({{\mathbb {R}}}^2{\setminus }\{0\})$ is homogeneous of degree $\lambda \in {{\mathbb {C}}},$ then the structure theorem (see [6, Sect. 2.5]) implies that $\sigma$ can be extended to ${{\mathbb {R}}}^2$ and can be represented in the form $\sigma =F\cdot r^\lambda$ where the so-called “characteristic” F belongs to $\mathcal D'({{\mathbb {S}}}^1)$ and

$$\begin{aligned} \langle \phi , F\cdot r^\lambda \rangle ={}_{\mathcal S({{\mathbb {R}}}^1)}\big \langle \,{}_{\mathcal D({{\mathbb {S}}}^1)} \langle \phi (t\omega ), F(\omega )\rangle _{\mathcal D'({{\mathbb {S}}}^1)}, t_+^{\lambda +1}\big \rangle _{\mathcal S'({{\mathbb {R}}}^1)},\quad \phi \in \mathcal S({{\mathbb {R}}}^2). \end{aligned}$$

If we identify ${{\mathbb {S}}}^1\subset {{\mathbb {R}}}^2\simeq {{\mathbb {C}}},$ then

$$\begin{aligned} \mathcal D'({{\mathbb {S}}}^1)\overset{\sim }{\longrightarrow }s'({{\mathbb {Z}}}): F\longmapsto (c_k)_{k\in {{\mathbb {Z}}}} \end{aligned}$$

where $c_k=\frac{1}{2\pi }\,\langle \omega ^{-k},F\rangle \in {{\mathbb {C}}}$ for $k\in {{\mathbb {Z}}}$ and $\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k$ converges to $F(\omega )$ in $\mathcal D'({{\mathbb {S}}}^1).$ The Hilbert spaces

$$\begin{aligned} \mathcal H^{(\mu )}({{\mathbb {S}}}^1)=\Bigl \{F(\omega )=\sum _{k\in {{\mathbb {Z}}}}c_k\omega ^k\in \mathcal D'({{\mathbb {S}}}^1);\, (c_k(1+|k|)^\mu )_{k\in {{\mathbb {Z}}}}\in l^2({{\mathbb {Z}}})\Bigr \},\quad \mu \in {{\mathbb {R}}}, \end{aligned}$$

are the usual Sobolev spaces on ${{\mathbb {S}}}^1.$

The distribution-valued function $z\mapsto t_+^z$ is meromorphic with simple poles at $z=-k,$$k\in {{\mathbb {N}}}.$ We have $\mathrm{Res}_{z=-k} t_+^z=(-1)^{k-1}(k-1)!^{-1} \delta ^{(k-1)}$ and we set

$$\begin{aligned} t_+^{-k}:=\mathop {\mathrm{Pf}}\limits _{z=-k} t_+^z= \lim _{z\rightarrow -k}\Bigl ( t^z-(z+k)^{-1} \mathop {\mathrm{Res}}\limits _{\lambda =-k} t^\lambda \Bigr ),\quad k\in {{\mathbb {N}}}. \end{aligned}$$

For $\lambda \in {{\mathbb {C}}}{\setminus }(-2-{{\mathbb {N}}}_0),$ this implies that $F\cdot r^\lambda$ is homogeneous not only in ${{\mathbb {R}}}^2{\setminus }\{0\},$ but also on ${{\mathbb {R}}}^2,$ whereas for $\lambda =-2-j,$$j\in {{\mathbb {N}}}_0,$$F\cdot r^\lambda$ is homogeneous on ${{\mathbb {R}}}^2$ if and only if the Fourier coefficients $c_k=(2\pi )^{-1}\langle \omega ^{-k},F\rangle$ vanish for $|k|\le j,$$k\in {{\mathbb {Z}}},$ see [6, Thm. 2.5.1, p. 58].

Furthermore, if $\mathcal S'_\lambda ({{\mathbb {R}}}^2),$$\lambda \in {{\mathbb {C}}},$ denotes the closed subspace of $\mathcal S'({{\mathbb {R}}}^2)$ of distributions which are homogeneous of degree $\lambda$ in ${{\mathbb {R}}}^2,$ then $\mathcal F:\mathcal S'_\lambda ({{\mathbb {R}}}^2)\overset{\sim }{\rightarrow }\mathcal S'_{-2-\lambda }({{\mathbb {R}}}^2)$ is an isomorphism of locally convex spaces. More explicitly, if $\lambda \in {{\mathbb {C}}}{\setminus }{{\mathbb {Z}}},$ then $\mathcal F(F\cdot r^\lambda )=G\cdot r^{-2-\lambda }$ where $F=\sum _{k\in {{\mathbb {Z}}}}c_k\omega ^k,$$G=\sum _{k\in {{\mathbb {Z}}}}d_k\omega ^k$ and

$$\begin{aligned} d_k=2^{2+\lambda }\pi \,\text {i}^{k}\frac{\Gamma (1+\frac{\lambda -k}{2})}{\Gamma (-\frac{\lambda +k}{2})}\, c_k,\quad k\in {{\mathbb {Z}}}. \end{aligned}$$

(2.2)

Formula (2.2) is a special case of the Bochner–Hecke formula, see [10, Ch. IV, Thm. 4.1, p. 160]. For a direct verification, note that, with $z=x+\text {i} y=r\omega ,$$r=|z|,$ and using [7, Equ. (1.6.18), p. 99], we have

$$\begin{aligned} \begin{aligned} \mathcal F(\omega ^k\cdot r^\lambda )&=\mathcal F( z^k\cdot r^{\lambda -k})=\text {i}^{|k|} (\partial _x+\mathrm{sign}(k)\text {i}\,\partial _y)^{|k|}\mathcal F(r^{\lambda -|k|}) \\&=2^{2+\lambda }\pi \,\text {i}^{|k|}\Gamma \big (1+\tfrac{\lambda -|k|}{2}\big )\bigl (\tfrac{1}{2}(\partial _x+\mathrm{sign}(k)\text {i}\,\partial _y)\bigr )^{|k|} \Bigl (\frac{r^{|k|-2-\lambda }}{\Gamma (\frac{|k|-\lambda }{2})}\Bigr ) \\&=2^{2+\lambda }\pi \,\text {i}^{|k|}\frac{\Gamma \big (1+\frac{\lambda -|k|}{2}\big )}{\Gamma \big (-\frac{\lambda +|k|}{2}\big )}\,\omega ^k\cdot r^{-2-\lambda }, \quad k\in {{\mathbb {Z}}}. \end{aligned} \end{aligned}$$

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 $\sigma \in \mathcal S'({{\mathbb {R}}}^2)$ can have a bounded Weyl transform $W_\sigma$ only under the condition that the degree $\lambda$ of homogeneity fulfills $-2\le \text {Re}\,\lambda \le 0.$ For homogeneous and radially symmetric symbols $\sigma =r^\lambda \in \mathcal S'({{\mathbb {R}}}^2),$ we have shown in [8, Sect. 5, Ex. 1] that $W_\sigma$ is bounded on $L^2({{\mathbb {R}}}^1)$ if and only if $-2\le \text {Re}\,\lambda \le 0,$$\lambda \ne -2.$ (Note that $r^{-2}=\mathrm{Pf}_{\lambda =-2} r^\lambda$ is not homogeneous on ${{\mathbb {R}}}^2.$ We remark, incidentally, that $W_\sigma$ is unbounded for $\sigma =r^{-2}\in \mathcal S'({{\mathbb {R}}}^2),$ see the remark to Proposition 3.)

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], $W_\sigma$ is bounded and even compact as an operator on $L^2({{\mathbb {R}}}^n)$ if $\sigma$ or $\mathcal F\sigma$ belong to $\cup _{p\in [1,2]} L^p({{\mathbb {R}}}^{2n}).$ In particular, if $\sigma =F(\omega )\cdot r^\lambda \in \mathcal S'({{\mathbb {R}}}^2)$ with $F\in L^2({{\mathbb {S}}}^1)$ and $-2<\text {Re}\,\lambda <-1,$ then $\sigma \in L^1({{\mathbb {R}}}^2)+ L^2({{\mathbb {R}}}^2)$ and hence $W_\sigma \in \text {Com}(L^2({{\mathbb {R}}}^1)).$ Similarly, if $-1<\mu =\text {Re}\,\lambda <0,$ and $F\in \mathcal H^{(\mu +1)}({{\mathbb {S}}}^1),$ i.e., $F(\omega )=\sum _{k\in {{\mathbb {Z}}}}c_k \omega ^k$ such that $(c_k|k|^{\mu +1})_{k\in {{\mathbb {Z}}}}\in l^2({{\mathbb {Z}}}),$ then we employ formula (2.2) and conclude once again that $W_\sigma \in \text {Com}(L^2({{\mathbb {R}}}^1))$ from the asymptotic behaviour

$$\begin{aligned} \frac{\Gamma (1+\frac{\lambda -|k|}{2})}{\Gamma (-\frac{\lambda +|k|}{2})}=(-1)^k\,\frac{\Gamma (1+\frac{|k|+\lambda }{2})}{\Gamma (\frac{|k|-\lambda }{2})} \sim (-1)^k\Bigl (\frac{|k|}{2}\Bigr )^{\lambda +1},\quad |k|\rightarrow \infty , \end{aligned}$$

see [5, Equ. 8.328.2]. We shall deduce more general results in Sect. 6 below.

3 The matrix representation of $W_\sigma$

The normalized Hermite functions

$$\begin{aligned} e_j=h_j/\Vert h_j\Vert _2=(2^j j! \sqrt{\pi })^{-1/2} h_j\in \mathcal S({{\mathbb {R}}}),\quad j\in {{\mathbb {N}}}_0, \end{aligned}$$

(3.1)

are an orthonormal basis in $L^2({{\mathbb {R}}}),$ see, e.g., [11, Thm. 19.3, p. 94]. This yields an isomorphism of Hilbert spaces:

$$\begin{aligned} L^2({{\mathbb {R}}})\overset{\sim }{\longrightarrow }l^2({{\mathbb {N}}}_0):f\longmapsto \Bigl (\int _{{{\mathbb {R}}}}f(x)e_j(x)\,\text {d}x\Bigr )_{j\in {{\mathbb {N}}}_0}. \end{aligned}$$

As was observed already by Schwartz, cf. [9, Ch. VII, Sect. 7, Ex. 7, pp. 260, 262], the sequence $e_j,$$j\in {{\mathbb {N}}}_0,$ also serves to identify the spaces $\mathcal S({{\mathbb {R}}})$ and $\mathcal S'({{\mathbb {R}}})$ with the sequence spaces $s({{\mathbb {N}}}_0)$ and $s'({{\mathbb {N}}}_0),$ respectively. Hence we obtain a commutative diagram:

$$\begin{aligned} \begin{array}{ccccc} \mathcal S({{\mathbb {R}}})\, &{}\hookrightarrow \, &{} L^2({{\mathbb {R}}})\, &{}\hookrightarrow \,&{}\mathcal S'({{\mathbb {R}}}) \\ \downarrow \!\!\wr &{} &{} \downarrow \!\!\wr &{}&{}\downarrow \!\!\wr \\ s({{\mathbb {N}}}_0)\,&{}\hookrightarrow \,&{} l^2({{\mathbb {N}}}_0)\, &{}\hookrightarrow \,&{}s'({{\mathbb {N}}}_0)\\ \end{array} \end{aligned}$$

Here the vertical isomorphisms in the diagram are all given by $T\mapsto (\langle e_j,T\rangle )_{j\in {{\mathbb {N}}}_0}.$

By the same token, using $\mathcal S'({{\mathbb {R}}}^2)\overset{\sim }{\rightarrow }s'({{\mathbb {N}}}_0^2),$ the mapping $W_\sigma :\mathcal S({{\mathbb {R}}})\rightarrow \mathcal S'({{\mathbb {R}}})$ is represented by the matrix $B_\sigma =(b_{jk})_{j,k\in {{\mathbb {N}}}_0}\in s'({{\mathbb {N}}}_0^2)$ where $W_\sigma e_k=\sum _{j\in {{\mathbb {N}}}_0} e_j b_{jk}$ for $k\in {{\mathbb {N}}}_0$ and the sums converge in $\mathcal S'({{\mathbb {R}}}).$ The matrix coefficients $b_{jk}$ of $W_\sigma$ are given by

$$\begin{aligned} b_{jk}=\langle e_j, W_\sigma e_k\rangle =\langle w(e_k,e_j),\sigma \rangle ,\quad j,k\in {{\mathbb {N}}}_0, \end{aligned}$$

(3.2)

on account of formula (2.1). Therefore, the operator $W_\sigma$ is bounded on $L^2({{\mathbb {R}}})$ if and only if

$$\begin{aligned} \exists C>0:\forall \phi \in s({{\mathbb {N}}}_0):\Big \Vert \Bigl (\sum _{k=0}^\infty b_{jk}\phi _k\Bigr )_{j\in {{\mathbb {N}}}_0}\Big \Vert _{l^2}\le C\Vert \phi \Vert _{l^2} \end{aligned}$$

holds.

Let us specialize now to distributions $\sigma \in \mathcal S'({{\mathbb {R}}}^2)$ that are tensor products in polar coordinates. For the radial dependency of $\sigma ,$ we take $R\in \mathcal S'([0,\infty )),$ which is the dual space of $\mathcal S\bigl ([0,\infty )\bigr )=\{\phi |_{[0,\infty )}:\phi \in \mathcal S({{\mathbb {R}}})\}$ equipped with the seminorms

$$\begin{aligned} \Vert \phi \Vert _{k,l}=\Vert (1+x)^k\phi ^{(l)}(x)\Vert _{L^\infty ([0,\infty ))},\quad k,l\in {{\mathbb {N}}}_0. \end{aligned}$$

As in [8, p. 776], we observe that

$$\begin{aligned} \mathcal S'\bigl ([0,\infty )\bigr )\overset{\sim }{\longrightarrow }\{S\in \mathcal S'({{\mathbb {R}}});\, \text {supp}\,S\subset [0,\infty )\}:S\longmapsto (\phi \mapsto \langle \phi |_{[0,\infty )},S\rangle ) \end{aligned}$$

is an isomorphism of locally convex spaces.

If, furthermore, $F\in \mathcal D'({{\mathbb {S}}}^1),$ then we define $F\cdot R\in \mathcal S'({{\mathbb {R}}}^2)$ by the following:

$$\begin{aligned} \langle \phi ,F\cdot R\rangle ={}_{\mathcal S([0,\infty ))}\big \langle r\cdot \langle \phi (r\omega ), F(\omega )\rangle , R(r)\big \rangle _{\mathcal S'([0,\infty ))},\quad \phi \in \mathcal S({{\mathbb {R}}}^2). \end{aligned}$$

(3.3)

Note that this is well-defined since the function

$$\begin{aligned} {{\mathbb {R}}}\longrightarrow {{\mathbb {C}}}:r\longmapsto r\cdot \langle \phi (r\omega ), F(\omega )\rangle \end{aligned}$$

belongs to $\mathcal S({{\mathbb {R}}}^1),$ but that the distribution $R\in \mathcal S'([0,\infty ))$ is not uniquely determined up to a factor by the distribution $\sigma =F\cdot R$ since, e.g., $F\cdot \delta =0.$

In the following Proposition 1, we state formulas for the matrix coefficients of $W_\sigma$ with respect to the normalized Hermite functions $e_j,e_k$ if $\sigma =F\cdot R\in \mathcal S'({{\mathbb {R}}}^2)$ 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 $L^\alpha _k$ be the Laguerre polynomials as in Sect. 1. Let $F(\omega )=\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k\in \mathcal D'({{\mathbb {S}}}^1)$ and $R\in \mathcal S'([0,\infty ))$ and let $\sigma =F\cdot R\in \mathcal S'({{\mathbb {R}}}^2)$ be defined as in formula (3.3). Let $W_\sigma e_k=\sum _{j\in {{\mathbb {N}}}_0} e_jb_{jk},$$k\in {{\mathbb {N}}}_0,$ with $(b_{jk})_{j,k\in {{\mathbb {N}}}_0}\in s'({{\mathbb {N}}}_0^2)$ being the matrix representation of $W_\sigma$ and define the symmetric matrix $(a_{jk})_{j,k\in {{\mathbb {N}}}_0}\in s'({{\mathbb {N}}}_0^2)$ by

$$\begin{aligned} a_{jk}=2(-1)^k\sqrt{\frac{k!}{j!}}\, 2^{(j-k)/2}\,{}_{\mathcal S([0,\infty ))} \langle r^{j-k+1} L^{j-k}_k(2r^2)\, {\mathrm{e}}^{-r^2},R(r)\rangle _{\mathcal S'([0,\infty ))}, \,\, j\ge k,\nonumber \\ \end{aligned}$$

(3.4)

and $a_{jk}=a_{kj}$ for $j\le k,$$j,k\in {{\mathbb {N}}}_0.$

Then the following holds:

• (a) For $j,k\in {{\mathbb {N}}}_0$ we have $b_{jk}=a_{jk} c_{k-j};$

• (b) if $\mu \in {{\mathbb {R}}}$ and

$$\begin{aligned} \exists C>0:\forall j,k\in {{\mathbb {N}}}_0:|a_{jk}|\le C\cdot (1+|j-k|)^\mu \end{aligned}$$

(3.5)

and $\bigl ((1+|k|)^\mu c_k\bigr )_{k\in {{\mathbb {Z}}}}\in l^1({{\mathbb {Z}}}),$ then $W_\sigma$ is bounded, i.e., $W_\sigma \in \mathcal L(L^2({{\mathbb {R}}})).$

Proof

(a) The formula for $b_{jk}$ in the proposition directly follows from the calculation of the Wigner transforms $w(e_j,e_k),$$j,k\in {{\mathbb {N}}}_0.$ This calculation goes back to C. Itzykson, W. Miller, and N.Ya. Vilenkin, see the historical comment in [3, p. 65]. Taking account of the different notations in the literature, we have, for $j,k\in {{\mathbb {N}}}_0$ with $j\ge k,$

$$\begin{aligned} (w(e_j,e_k))(r\omega )=\frac{2(-1)^k}{2\pi }\sqrt{\frac{k!}{j!}}\, 2^{(j-k)/2}\, r^{j-k}\omega ^{k-j}\, L^{j-k}_k(2r^2)\, \text {e}^{-r^2}, \end{aligned}$$

(3.6)

where $r>0$ and $\omega \in {{\mathbb {S}}}^1,$ see [11, Thm. 24.1, p. 113], [3, Thm. 1.104, p. 64]. Furthermore we have, quite trivially, $w(e_k,e_j)=\overline{w(e_j,e_k)}$ for $j,k\in {{\mathbb {N}}}_0.$

Hence employing formulas (3.2), (3.3), (3.4) and (3.6), we obtain

$$\begin{aligned} b_{jk}=\langle w(e_k,e_j),\sigma \rangle =\big \langle r\cdot \langle \,\overline{w(e_j,e_k)(r\omega )}\,, F(\omega )\rangle , R(r)\big \rangle = a_{jk}c_{k-j},\quad j,k\in {{\mathbb {N}}}_0. \end{aligned}$$

(b) In the second part, we will closely follow the proof of Theorem 27.3 in [11, p. 129]. Assuming condition (3.5) and $\bigl ((1+|k|)^\mu c_k\bigr )_{k\in {{\mathbb {Z}}}}\in l^1({{\mathbb {Z}}}),$ we set $d_k=C(1+|k|)^\mu |c_k|,$$k\in {{\mathbb {Z}}}.$ Then the sequence $(d_k)_{k\in {{\mathbb {Z}}}}$ belongs to $l^1({{\mathbb {Z}}})$ and

$$\begin{aligned} |b_{jk}|=|a_{jk}|\cdot |c_{k-j}|\le C\cdot (1+|k-j|)^\mu |c_{k-j}|= d_{k-j},\quad j,k\in {{\mathbb {N}}}_0. \end{aligned}$$

Hence, if $\phi \in s({{\mathbb {N}}}_0)$ and if we set $\phi _k=0$ for $k<0,$ then the matrix $B_\sigma =(b_{jk})_{j,k\in {{\mathbb {N}}}_0}$ of $W_\sigma$ fulfills

$$\begin{aligned} \begin{aligned} \Vert B_\sigma \phi \Vert _{l^2({{\mathbb {N}}}_0)}&=\Big \Vert \Bigl (\sum _{k=0}^\infty b_{jk}\phi _k\Bigr )_{j\in {{\mathbb {N}}}_0}\Big \Vert _{l^2({{\mathbb {N}}}_0)} \le \Big \Vert \Bigl (\sum _{k\in {{\mathbb {Z}}}} d_{k-j}|\phi _k|\Bigr )_{j\in {{\mathbb {N}}}_0}\Big \Vert _{l^2({{\mathbb {N}}}_0)} \\&=\Big \Vert \Bigl (\sum _{l\in {{\mathbb {Z}}}}d_l|\phi _{j+l}|\Bigr )_{j\in {{\mathbb {N}}}_0}\Big \Vert _{l^2({{\mathbb {N}}}_0)} \le \Vert \phi \Vert _{l^2({{\mathbb {N}}}_0)}\sum _{l\in {{\mathbb {Z}}}}d_l, \end{aligned} \end{aligned}$$

i.e., $B_\sigma$ and thus also $W_\sigma$ are bounded. This completes the proof. $\square$

Remarks (1) Let us first discuss the special case of $\sigma \in \mathcal S'({{\mathbb {R}}}^2)$ being radially symmetric. We can set then $F=1$ and hence $\sigma$ is given by $R(r)\in \mathcal S'([0,\infty )).$ As we have mentioned above, R is not uniquely determined by $\sigma .$ However, as explained in [8, p. 776], there is a unique $S\in \mathcal S'([0,\infty ))$ such that $\sigma$ is represented by the formula

$$\begin{aligned} \langle \phi ,\sigma \rangle ={}_{\mathcal S([0,\infty ))}\left\langle \tfrac{1}{2}\int _{{{\mathbb {S}}}^1}\phi (\sqrt{t}\,\omega )\,\text {d}\omega , S(t)\right\rangle _{\mathcal S'([0,\infty ))}, \quad \phi \in \mathcal S({{\mathbb {R}}}^2). \end{aligned}$$

(3.7)

This means that $S\in \mathcal S'([0,\infty ))$ satisfies

$$\begin{aligned} \langle 2r\psi (r^2),R(r)\rangle =\langle \psi ,S\rangle , \quad \psi \in \mathcal S([0,\infty )). \end{aligned}$$

If we set $F=1$ in Proposition 1, we obtain $c_k=\delta _{k0}$ and hence the matrix $B_\sigma$ is diagonal with the eigenvalues

$$\begin{aligned} \lambda _j=a_{jj}=2(-1)^j\langle r L^0_j(2r^2)\,\text {e}^{-r^2}, R(r)\rangle =(-1)^j\langle L_j^0(2t)\,\text {e}^{-t},S(t)\rangle , \quad j\in {{\mathbb {N}}}_0.\nonumber \\ \end{aligned}$$

(3.8)

Formula (3.8) coincides with [8, Prop. 3, Equ. (3.5), p. 777].

However note that condition (3.5) in Proposition 1 above is overly restrictive in the case of radially symmetric distributions since, clearly, the diagonal matrix $B_\sigma$ is bounded on $l^2({{\mathbb {N}}}_0)$ if and only if the sequence of eigenvalues $(a_{jj})_{j\in {{\mathbb {N}}}_0}$ is bounded. Hence no restriction on $a_{jk},$$j\ne k,$ is necessary. More generally, if F is a trigonometric polynomial, i.e., if $F(\omega )=\sum _{k\in M}c_k\omega ^k$ where M is a finite subset of ${{\mathbb {Z}}}$ with $c_k\ne 0$ for $k\in M$ and if $\sigma =F\cdot R,$ then $W_\sigma$ is bounded on $L^2({{\mathbb {R}}})$ if and only if the set $\{a_{jk};\, k-j\in M\}$ is bounded in ${{\mathbb {C}}}.$

(2) In [8, Prop. 4, Equ. (4.2), p. 779], a representation of the eigenvalues $\lambda _j=a_{jj},$$j\in {{\mathbb {N}}}_0,$ by integrals with rational kernels was derived for $\sigma \in \mathcal S'({{\mathbb {R}}}^2)$ radially symmetric and such that $\sigma (x,y)=(\mathcal FT)(x^2+y^2)$ with $T\in \mathcal D'_{L^1,-1}({{\mathbb {R}}}).$ This formula, i.e.,

$$\begin{aligned} \lambda _j={}_{\mathcal D_{L^\infty ,1}}\!\left\langle \frac{(1-\text {i} t)^j}{(1+\text {i} t)^{j+1}}, T(t)\right\rangle _{\mathcal D'_{L^1,-1}},\quad j\in {{\mathbb {N}}}_0, \end{aligned}$$

was useful in some examples in calculating the asymptotic behavior of $\lambda _j$ for $j\rightarrow \infty ,$ 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\in {{\mathbb {N}}}_0,$$j\ge k,$$T\in \mathcal D'_{L^1,-1}({{\mathbb {R}}})$ and $R(r)=(\mathcal FT)(r^2)\cdot r^{j-k}$ holds in $\mathcal S'([0,\infty )),$ then

$$\begin{aligned} a_{jk}=\sqrt{\frac{j!}{k!}}\, 2^{(j-k)/2}\,{}_{\mathcal D_{L^\infty ,1}}\!\big \langle \frac{(1-\text {i} t)^k}{(1+\text {i} t)^{j+1}}, T(t)\rangle _{\mathcal D'_{L^1,-1}}. \end{aligned}$$

(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].

4 The case of homogeneous symbols of degree 0

We specialize now our investigation to homogeneous symbols $\sigma =F\cdot r^\lambda ,$$\lambda \in {{\mathbb {C}}},$$F(\omega )=\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k\in \mathcal D'({{\mathbb {S}}}^1).$ We assume that $\sigma$ is homogeneous on ${{\mathbb {R}}}^2$ and hence that $c_k=0$ for $|k|\le j$ if $\lambda =-2-j,$$j\in {{\mathbb {N}}}_0,$ cf. Sect. 2.

In order to calculate $a_{jk}(\lambda )$ for $R(r)=r_+^\lambda \in \mathcal S'([0,\infty ))$ according to formula (3.4), we first assume that $\lambda \in {{\mathbb {C}}}{\setminus }{{\mathbb {Z}}}$ with $\text {Re}\,\lambda >-2.$ Then [1, Equ. 11.44, p. 110] and [5, Equ. 9.131.1] yield the following for $j,k\in {{\mathbb {N}}}_0$ with $j\ge k:$

$$\begin{aligned} a_{jk}(\lambda )= & {} 2(-1)^k\sqrt{\frac{k!}{j!}}\, 2^{(j-k)/2} \int _0^\infty r^{\lambda +j-k+1} L^{j-k}_k(2r^2)\, \text {e}^{-r^2}\,\text {d}r \nonumber \\= & {} \frac{(-1)^k}{2^{\lambda /2+1}}\sqrt{\frac{k!}{j!}} \int _0^\infty t^{(\lambda +j-k)/2} L^{j-k}_k(t)\, \text {e}^{-t/2}\,\text {d}t \nonumber \\= & {} 2^{(j-k)/2}\frac{\Gamma (\frac{\lambda +j+k}{2}+1)}{\sqrt{j!\,k!}}\cdot {}_2F_1\bigl (-k,\tfrac{j-k-\lambda }{2};-\tfrac{j+k+\lambda }{2};-1\bigr ) \end{aligned}$$

(4.1)

$$\begin{aligned}= & {} 2^{(j+k)/2}\frac{\Gamma (\frac{\lambda +j+k}{2}+1)}{\sqrt{j!\,k!}}\cdot {}_2F_1\bigl (-j,-k;-\tfrac{j+k+\lambda }{2};\tfrac{1}{2}\bigr ) \end{aligned}$$

(4.2)

By analytic continuation, formulas (4.1), (4.2) hold for $\lambda \in {{\mathbb {C}}}{\setminus }{{\mathbb {Z}}}$ and, due to the symmetry of (4.2) in j, k,  for all $j,k\in {{\mathbb {N}}}_0.$ But note that the hypergeometric function ${}_2 F_1(a,b;c;z)$ is defined for $c\in -{{\mathbb {N}}}_0$ only if $[a\in -{{\mathbb {N}}}_0$ and $a>c]$ or $[b\in -{{\mathbb {N}}}_0$ and $b>c].$ Because of that and because of the gamma factor in (4.1), (4.2), we have to exercise some care if $\lambda \in {{\mathbb {Z}}}.$

Let us now specialize further and consider the case $\lambda =0,$ i.e., the case of homogeneous symbols $\sigma =F(\omega )$ of degree 0.

Proposition 2

Let $F(\omega )=\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k\in \mathcal D'({{\mathbb {S}}}^1)$ and $\sigma =F\cdot 1\in \mathcal S'({{\mathbb {R}}}^2)$ be homogeneous of degree 0. Then the following holds:

(a) If $\alpha _j=\sqrt{\Gamma (\frac{j}{2}+1)/\Gamma (\frac{j+1}{2})}$ for $j\in {{\mathbb {N}}}_0$ and $a_{jk}=a_{jk}(0)$ is as in Proposition 1 for $R(r)=1,$ then

$$\begin{aligned} a_{jk}(0)=\left\{ \begin{array}{lll} &{}\frac{\alpha _{\max (j,k)}}{\alpha _{\min (j,k)}} &{}:\min (j,k)\text { is even},\\ &{}\frac{\alpha _{\min (j,k)}}{\alpha _{\max (j,k)}} &{}:\min (j,k)\text { is odd} \end{array} \right\} ,\quad j,k\in {{\mathbb {N}}}_0. \end{aligned}$$

(4.3)

(b) If $(\root 4 \of {|k|}\, c_k)_{k\in {{\mathbb {Z}}}}\in l^1({{\mathbb {Z}}}),$ then $W_\sigma \in \mathcal L(L^2({{\mathbb {R}}})).$

Proof

(a) If $j=k\in {{\mathbb {N}}}_0,$ then formula (4.1) yields $a_{jj}(0)=1.$ If $j\ne k,$ then formula (4.2) is valid. Let us first assume $j>k$ and k even. For $\text {Re}\, z<-k,$ [2, Equ. 7.3.7.4] yields

$$\begin{aligned} {}_2F_1\bigl (z,-k;\tfrac{z-k}{2};\tfrac{1}{2}\bigr )=\sqrt{\pi }\,\frac{\Gamma (\frac{z-k}{2})}{\Gamma (\frac{z}{2})\Gamma (\frac{1-k}{2})}= \frac{\Gamma (1-\frac{z}{2})\Gamma (\frac{k+1}{2})}{\sqrt{\pi }\,\Gamma (\frac{k-z}{2}+1)} \end{aligned}$$

and hence

$$\begin{aligned} a_{jk}(0)=2^{(j+k)/2}\frac{\Gamma (\frac{j}{2}+1)\Gamma (\frac{k+1}{2})}{\sqrt{\pi \,\Gamma (j+1)\Gamma (k+1)}}=\frac{\alpha _j}{\alpha _k} \end{aligned}$$

on account of the complement formula and of the duplication formula for the gamma function.

Similarly, if k is odd and $\text {Re}\,z<-k,$ then [2, Equ. 7.3.7.4] furnishes

$$\begin{aligned} {}_2F_1\bigl (z,-k;\tfrac{z-k}{2};\tfrac{1}{2}\bigr )=\sqrt{\pi }\,\frac{\Gamma (\frac{z-k}{2})}{\Gamma (\frac{ z+1}{2})\Gamma (-\frac{k}{2})}= \frac{\Gamma (\frac{1-z}{2})\Gamma (\frac{k}{2}+1)}{\sqrt{\pi }\,\Gamma (\frac{k-z}{2}+1)} \end{aligned}$$

and hence $a_{jk}(0)=\alpha _k/\alpha _j$ for $j>k.$ This proves formula (4.3).

(b) Let us next estimate $a_{jk}(0).$ Using Euler’s psi function $\psi (x)=(\log \Gamma (x))'$ for $x>0$ we obtain

$$\begin{aligned} \log \alpha _j=\frac{1}{2}\int _{(j+1)/2}^{j/2+1}\psi (x)\,\text {d}x. \end{aligned}$$

(4.4)

Equation (4.4) implies, incidentally, that $\alpha _j$ is monotonically increasing since the same holds for the function $\psi$ due to the logarithmic convexity of the gamma function, see, e.g., [5, Equ. 8.363.3].

Since $C_1+\log x\le \psi (x)\le C_2+\log x$ holds for all $x\ge \frac{1}{2}$ and appropriate constants $C_1,C_2\in {{\mathbb {R}}},$ we conclude that

$$\begin{aligned} \exists C>0:\forall j\in {{\mathbb {N}}}_0: C^{-1}\root 4 \of {j+1}\le \alpha _j\le C\root 4 \of {j+1}. \end{aligned}$$

(4.5)

This implies

$$\begin{aligned} \exists C&>0:\forall j,k\in {{\mathbb {N}}}_0\text { with }j\ge k: 0\le a_{jk}(0)\le \frac{\alpha _j}{\alpha _k}\nonumber \\&\le C\root 4 \of {\frac{j+1}{k+1}} =C\Bigl (1+\frac{j-k}{k+1}\Bigr )^{1/4}. \end{aligned}$$

(4.6)

Therefore, condition (3.5) in Proposition 1 holds for $\mu =\frac{1}{4}$ and this implies that $W_\sigma$ is bounded on $L^2({{\mathbb {R}}})$ if the condition $(\root 4 \of {|k|}\, c_k)_{k\in {{\mathbb {Z}}}}\in l^1({{\mathbb {Z}}})$ is assumed. The proof is complete. $\square$

Remark

The matrix $(a_{jk}(\lambda ))_{j,k\in {{\mathbb {N}}}_0}$ can be calculated similarly for each integral $\lambda .$ E.g., if $\lambda =-1,$ then [2, Equ. 7.3.7.5] implies that $a_{jk}(-1)=(\alpha _j\alpha _k)^{-1}$ if $\min (j,k)$ is even and $a_{jk}(-1)$ vanishes if $\min (j,k)$ is odd. Hence

$$\begin{aligned} \exists C>0:\forall j,k\in {{\mathbb {N}}}_0: 0\le a_{jk}(-1)\le C(1+|j-k|)^{-1/4} \end{aligned}$$

and $W_\sigma \in \mathcal L(L^2({{\mathbb {R}}}))$ for $\sigma =F\cdot r^{-1}$ if $\sum _{k\in {{\mathbb {Z}}}}|c_k|(1+|k|)^{-1/4}<\infty .$

5 Necessary conditions

We shall prove here that $\sigma \in \mathcal S'({{\mathbb {R}}}^2)$ homogeneous of degree $\lambda$ can have a bounded Weyl transform only if the real part of $\lambda$ lies in the interval $[-2,0]$ and if the characteristic F of $\sigma$ belongs to the Sobolev space $\mathcal H^{(1/4+\mu /2)}({{\mathbb {S}}}^1).$ In deriving this result, we need yet another representation of $a_{jk}(\lambda )$ 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 $\lambda \in {{\mathbb {C}}}{\setminus }{{\mathbb {Z}}}$ and $j,k\in {{\mathbb {N}}}_0$ with $j\ge k:$

$$\begin{aligned} a_{jk}(\lambda )=2^{(j-k)/2}\sqrt{\frac{j!}{k!}}\,\frac{\Gamma (\frac{\lambda +j-k}{2}+1)}{(j-k)!}\cdot {}_2F_1\bigl (-k,\tfrac{j-k-\lambda }{2};j-k+1;2\bigr ). \end{aligned}$$

(5.1)

Let us now digress a little bit and derive the “complement formula” for the matrix coefficients $a_{jk}(\lambda )$ from formula (5.1). Upon applying [5, Equ. 9.131.1] to formula (5.1) we obtain, still for $j\ge k,$

$$\begin{aligned} a_{jk}(\lambda )=2^{(j-k)/2}\sqrt{\frac{j!}{k!}}\,\frac{\Gamma (\frac{\lambda +j-k}{2}+1)}{(j-k)!}\cdot (-1)^k {}_2F_1\bigl (-k,\tfrac{j-k+\lambda +2}{2};j-k+1;2\bigr ), \end{aligned}$$

wherefrom we conclude that

$$\begin{aligned} a_{jk}(\lambda )=(-1)^{k}\,\frac{\Gamma (\frac{\lambda +j-k}{2}+1)}{\Gamma (\frac{-\lambda +j-k}{2})}\, a_{jk}(-2-\lambda ),\quad j,k\in {{\mathbb {N}}}_0,\ \lambda \in {{\mathbb {C}}}{\setminus }{{\mathbb {Z}}}. \end{aligned}$$

(5.2)

Note that the factor in (5.2) in front of $a_{jk}(-2-\lambda )$ 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 $W_\sigma$ and $W_{\mathcal F\sigma }$ are closely connected.

Proposition 3

Let $\sigma \in \mathcal S'({{\mathbb {R}}}^2)$ be homogeneous on ${{\mathbb {R}}}^2$ of degree $\lambda \in {{\mathbb {C}}}$ and such that $W_\sigma$ is bounded on $L^2({{\mathbb {R}}}).$ Then $-2\le {\mathrm{Re}}\,\lambda \le 0$ and [either $\sigma$ is a multiple of $\delta$ or $\sigma =F\cdot r^\lambda$ with $F\in \mathcal H^{(1/4+\mu /2)}({{\mathbb {S}}}^1)$ for $\mu ={\mathrm{Re}}\,\lambda$].

Proof

(a) Let us first settle the case of homogeneous $\sigma$ with support contained in $\{0\}.$ Then $\lambda =-2-j,$$j\in {{\mathbb {N}}}_0,$ and $\sigma$ is a linear combination of $\partial _x^k\partial _y^l\delta$ for $k,l\in {{\mathbb {N}}}_0$ with $k+l=j.$ This implies that $W_\sigma$ is a linear combination of the operators $A_{kl},$$k+l=j,$ where $(A_{kl}\phi )(x)=x^k\phi ^{(l)}(-x)$ for $\phi \in \mathcal S({{\mathbb {R}}}).$ Therefore, $W_\sigma$ is bounded in this case if and only if $j=0.$ Indeed, for $\phi \in \mathcal S({{\mathbb {R}}}),$ we have

$$\begin{aligned} \lim _{N\rightarrow \infty } N^{-k}\Vert A_{kl}(\phi (x+N))\Vert _2=\Vert \phi ^{(l)}\Vert _2 \end{aligned}$$

and the boundedness of $A_{kl}$ implies $k=0,$ and

$$\begin{aligned} \Vert A_{0l}(\phi (Nx))\Vert _2=N^{l-1/2}\Vert \phi ^{(l)}\Vert _2,\quad N>0, \end{aligned}$$

and the boundedness of $A_{0l}$ implies $l=0$ and hence $\sigma$ is a multiple of $\delta$.

(b) Let now $\sigma =F\cdot r^\lambda$ with $F(\omega )=\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k\in \mathcal D'({{\mathbb {S}}}^1)$ and assume that $c_m\ne 0$ for some $m\in {{\mathbb {Z}}}.$ In particular, in the case of $\lambda =-2-j,$$j\in {{\mathbb {N}}}_0,$ the homogeneity of $\sigma$ on ${{\mathbb {R}}}^2$ implies that $|m|>j.$

Since $W_\sigma \in \mathcal L(L^2({{\mathbb {R}}})),$ its matrix coefficients $b_{jk}=\langle e_j,W_\sigma e_k\rangle$ are bounded in modulus by $\Vert W_\sigma \Vert _{\mathcal L(L^2({{\mathbb {R}}}))}.$ Hence also the complex sequence $a_{k,k+m}(\lambda )=b_{k,k+m}/c_m,$$k\in {{\mathbb {N}}}_0,$$k\ge -m,$ must be bounded.

Let us next determine the asymptotic behavior of $a_{k+m,k}(\lambda )$ for $k\rightarrow \infty$ and $m\in {{\mathbb {N}}}_0$ fixed, and for $\lambda \in {{\mathbb {C}}}{\setminus }(-2-{{\mathbb {N}}}_0)$ or such that $\lambda =-2-j,$$j\in {{\mathbb {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

$$\begin{aligned} a_{k+m,k}(\lambda )=2^{m/2}\sqrt{\frac{(k+m)!}{k!}}\,\left\langle \frac{t_+^{(m-\lambda )/2-1}}{\Gamma (\frac{m-\lambda }{2})}\cdot (1-t)_+^{(m+\lambda )/2}, (1-2t)^k\right\rangle ,\,\, k,m\in {{\mathbb {N}}}_0.\nonumber \\ \end{aligned}$$

(5.3)

Note that $t_+^{(m-\lambda )/2-1}/\Gamma (\frac{m-\lambda }{2})$ is an entire distribution-valued function of $\lambda$ which can be multiplied with $(1-t)_+^{(m+\lambda )/2}$ since the singular supports of the two distributions are disjoint. Also note that $(1-t)_+^{(m+\lambda )/2}$ is analytic at the value of $\lambda$ which we consider.

We then obtain the asymptotic behavior of $a_{k+m,k}(\lambda )$ for $k\rightarrow \infty$ 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

$$\begin{aligned} \begin{aligned} a_{k+m,k}(\lambda )&\sim 2^{m/2}\sqrt{\frac{(k+m)!}{k!}} \left[ \frac{\Gamma (\frac{m+\lambda }{2}+2k+1)}{\Gamma (m+2k+1)}+ (-1)^k \frac{\Gamma (\frac{m-\lambda }{2}+2k)\Gamma (\frac{m+\lambda }{2}+1)}{\Gamma (\frac{m-\lambda }{2})\Gamma (m+2k+1)}\right] \\&\sim (2k)^{m/2}\biggl [(2k)^{(\lambda -m)/2}+ (-1)^k \frac{\Gamma (\frac{m+\lambda }{2}+1)}{\Gamma (\frac{m-\lambda }{2})}(2k)^{-(m+\lambda +2)/2}\biggr ]. \end{aligned} \end{aligned}$$

Hence, eventually,

$$\begin{aligned} a_{k+m,k}(\lambda )\sim \left\{ \begin{aligned}&(2k)^{\lambda /2}&:\text {Re}\,\lambda >-1,\\&(2k)^{\lambda /2}+(-1)^k\tfrac{\Gamma (\frac{m+\lambda }{2}+1)}{\Gamma (\frac{m-\lambda }{2})}\,(2k)^{-\lambda /2-1}&:\text {Re}\,\lambda =-1,\\&(-1)^k\tfrac{\Gamma (\frac{m+\lambda }{2}+1)}{\Gamma (\frac{m-\lambda }{2})}\,(2k)^{-\lambda /2-1}&:\text {Re}\,\lambda <-1, \end{aligned}\right. \end{aligned}$$

(5.4)

for $k\rightarrow \infty$ and $m\in {{\mathbb {N}}}_0$ with $m>|2+\lambda |$ if $\lambda \in -2-{{\mathbb {N}}}_0.$ Here the precise meaning of $v_k\sim w_k$ is $\lim _{k\rightarrow \infty }k^{-c}(v_k-w_k)=0$ for $c=\frac{1}{2}\max (\text {Re}\,\lambda ,-2-\text {Re}\,\lambda ).$ We remark that the special case of $m=0$ was shown already in [8, p. 783].

According to the above, if $c_m\ne 0$ for some $m\in {{\mathbb {N}}}_0,$ then the boundedness of $a_{k+m,k}(\lambda )$ implies $\text {Re}\,\lambda \in [-2,0]$ due to formula (5.4). If instead $c_m\ne 0$ for $m<0,$ then we use the boundedness of $a_{k,k+m}(\lambda )=a_{l+|m|,l}(\lambda )$ for $l=k+m$ with $k\ge |m|$.

(c) Let us finally show that $F\in \mathcal H^{(1/4+\mu /2)}({{\mathbb {S}}}^1)$ for $\mu =\text {Re}\,\lambda .$ First we observe that

$$\begin{aligned} a_{j0}(\lambda )=2^{j/2}\,\frac{\Gamma (\frac{\lambda +j}{2}+1)}{\sqrt{j!}}=\root 4 \of {\pi }\,\sqrt{\frac{\Gamma (\frac{\lambda +j}{2}+1)^2}{\Gamma (\frac{j+1}{2})\Gamma (\frac{j}{2}+1)}}\sim \root 4 \of {\pi }\,\Bigl (\frac{j}{2}\Bigr )^{1/4+\lambda /2},\quad j\rightarrow \infty , \end{aligned}$$

according to [5, Equ. 8.328.2]. If $e_j,$$j\in {{\mathbb {N}}}_0,$ denotes, as before, the orthonormal basis of normalized Hermite functions, see formula (3.1), we infer therefore from

$$\begin{aligned} W_\sigma e_0=\sum _{j=0}^\infty e_j b_{j0}=\sum _{j=0}^\infty e_j a_{j0}c_{-j} \end{aligned}$$

and from $W_\sigma \in \mathcal L(L^2({{\mathbb {R}}}))$ that $(a_{j0}c_{-j})_{j\in {{\mathbb {N}}}_0}\in l^2({{\mathbb {N}}}_0)$ and hence that also the sequence $(c_{-j}(1+j)^{1/4+\mu /2})_{j\in {{\mathbb {N}}}_0}$ belongs to $l^2({{\mathbb {N}}}_0).$

Similarly, if $\phi \in s({{\mathbb {N}}}_0)$ with $\Vert \phi \Vert _{l^2({{\mathbb {N}}}_0)}\le 1,$ then

$$\begin{aligned} \Vert W_\sigma \Vert _{\mathcal L(L^2({{\mathbb {R}}}))}\ge \Big |\Big \langle W_\sigma \Bigl (\sum _{k\in {{\mathbb {N}}}_0}\phi _k e_k\Bigr ),e_0\Big \rangle \Big | =\Big |\sum _{k=0}^\infty a_{0k}c_k\phi _k\Big | \end{aligned}$$

and thus $(c_k(1+k)^{1/4+\mu /2})_{k\in {{\mathbb {N}}}_0}\in l^2({{\mathbb {N}}}_0).$ Altogether we conclude that F belongs to the Sobolev space $\mathcal H^{(1/4+\mu /2)}({{\mathbb {S}}}^1)$ and this completes the proof. $\square$

Remark

Let us yet comment on the situation for symbols $\sigma$ that are homogeneous on ${{\mathbb {R}}}^2{\setminus }\{0\},$ but only associated homogeneous on ${{\mathbb {R}}}^2,$ i.e., $\sigma =F\cdot r^{-2-j},$$j\in {{\mathbb {N}}}_0,$ and $F(\omega )=\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k$ with $c_k\ne 0$ for some $k\in {{\mathbb {Z}}}$ with $|k|\le j.$ In this case, $W_\sigma$ cannot be bounded on $L^2({{\mathbb {R}}}).$ 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 $\lambda =-2.$

According to the complement formula (5.2), we obtain

$$\begin{aligned} a_{jk}(-2)=(-1)^{\min (j,k)}\cdot \frac{2 a_{jk}(0)}{|j-k|},\quad j\ne k\in {{\mathbb {N}}}_0. \end{aligned}$$

In particular, this implies

$$\begin{aligned} \exists C>0:\forall j\ne k\in {{\mathbb {N}}}_0:|a_{jk}(-2)|\le C(1+|j-k|)^{-3/4} \end{aligned}$$

and hence $W_\sigma \in \mathcal L(L^2({{\mathbb {R}}}))$ if $\sigma =F\cdot r^{-2}$, $F=\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k,$$\sigma$ is homogeneous on ${{\mathbb {R}}}^2,$ i.e., $c_0=0,$ and $\sum _{k\in {{\mathbb {Z}}}}|c_k|(1+|k|)^{-3/4}<\infty .$

On the other hand, $a_{jj}(-2)$ grows logarithmically for $j\rightarrow \infty$ and thus $W_\sigma$ is not bounded if $\sigma = r^{-2}=\mathrm{Pf}_{\lambda =-2}r^\lambda .$ In fact, setting $m=0$ and $k=j\in {{\mathbb {N}}}$ in formula (5.3) yields

$$\begin{aligned} a_{jj}(-2)&=\mathop {\mathrm{Pf}}\limits _{\lambda =-2}a_{jj}(\lambda )=\mathop {\mathrm{Pf}}\limits _{z=0} a_{jj}(-2-2z)\\&=\Big \langle \mathop {\mathrm{Pf}}\limits _{z=0}\Bigl (\frac{t_+^z}{\Gamma (1+z)}\cdot (1-t)_+^{-1-z}\Bigr ),(1-2t)^j\Big \rangle \\&=\Big \langle \frac{\text {d}}{\text {d}z} \Bigl (\frac{t_+^z}{\Gamma (1+z)}\left. \Bigr )\right| _{z=0}\cdot \mathop {\mathrm{Res}}\limits _{z=0} (1-t)_+^{-1-z}+Y(t)(1-t)_+^{-1},(1-2t)^j\Big \rangle \\&=\langle \bigl (\psi (1)-\log t\bigr )\cdot \delta (1-t)+Y(t)(1-t)_+^{-1},(1-2t)^j\rangle \\&=\psi (1)(-1)^j+\int _0^1\frac{(1-2t)^j-(-1)^j}{1-t}\,\text {d}t\\&=(-1)^j\Bigl [\psi (1)+2j\int _0^1(1-2s)^{j-1}\log s\,\text {d}s\Bigr ]\\&=(-1)^j\Bigl [\psi (1)+2j\int _0^1(1-s)^{2j-2}\log s\,\text {d}s+ o(1)\Bigr ] \\&=2(-1)^{j+1}\bigl [\psi (2j)-\psi (1)\bigr ]+o(1). \end{aligned}$$

Hence $a_{jj}(-2)=2(-1)^{j+1}\bigl [\log (2j)-\psi (1)\bigr ]+o(1)$ for $j\rightarrow \infty .$ (Here the terms o(1) denote series converging to 0).

6 A sufficient condition ensuring the boundedness of $W_\sigma$

When comparing the three examples $\lambda =0$ (in Proposition 2), $\lambda =-1$ (in the remark following Proposition 2) and $\lambda =-2$ (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 $\sigma = F\cdot r^\lambda$ in order to guarantee that $W_\sigma$ is bounded. The following proposition incorporates these special cases into a general statement.

Proposition 4

Let $F(\omega )=\sum _{k\in {{\mathbb {Z}}}} c_k\omega ^k\in \mathcal D'({{\mathbb {S}}}^1)$ and $\lambda \in {{\mathbb {C}}}$ such that $\mu ={\mathrm{Re}}\,\lambda \in [-2,0]$ and assume that $\sigma =F\cdot r^\lambda$ is homogeneous on ${{\mathbb {R}}}^2.$ (This just means that $c_0$ vanishes if $\lambda =-2.)$ Then the following holds:

If $\sum _{k\in {{\mathbb {Z}}}}|c_k|(1+|k|)^{1/4+\mu /2}<\infty ,$ then $W_\sigma$ is a bounded operator on $L^2({{\mathbb {R}}}).$

Proof

(a) We have dealt with the special case of $\lambda =-2$ in the remark following Proposition 3. We can therefore assume that $\lambda \ne -2.$ According to condition (3.5) in Proposition 1, we then have to verify the following estimate:

$$\begin{aligned} \exists C>0:\forall j,k\in {{\mathbb {N}}}_0:|a_{jk}(\lambda )|\le C(1+|j-k|)^{1/4+\mu /2}. \end{aligned}$$

(6.1)

Note that the asymptotic expansion in formula (5.4) already shows that $a_{jj}(\lambda ),$$j\in {{\mathbb {N}}}_0,$ is a bounded sequence. Therefore it suffices to prove the estimate in (6.1) for $j>k.$

(b) Let us define now

$$\begin{aligned} {\tilde{a}}_{jk}(\lambda ):=\int _0^\infty \frac{(1-s)^k}{(1+s)^{j+1}}\, s^{(j-k-\lambda )/2-1}\,\text {d}s,\quad j,k\in {{\mathbb {N}}}_0,\ j>k. \end{aligned}$$

(6.2)

We point out that the integral in (6.2) is absolutely convergent for $\mu =\text {Re}\,\lambda \in [-2,0].$

Due to [5, Equs. 3.197.1 and 9.131.1] the integral in (6.2) yields the following:

$$\begin{aligned} \tilde{a}_{jk}(\lambda )=\frac{\Gamma (\frac{j-k-\lambda }{2})\Gamma (\frac{\lambda +j-k}{2}+1)}{(j-k)!}\cdot {}_2F_1(-k,\tfrac{j-k-\lambda }{2};j-k+1,2). \end{aligned}$$

(6.3)

Combining formulas (5.1) and (6.3), we obtain

$$\begin{aligned} a_{jk}(\lambda )=2^{(j-k)/2}\sqrt{\frac{j!}{k!}}\cdot \frac{\tilde{a}_{jk}(\lambda )}{\Gamma (\frac{j-k-\lambda }{2})}, \quad j,k\in {{\mathbb {N}}}_0,\ j>k,\ \mu =\text {Re}\,\lambda \in [-2,0].\nonumber \\ \end{aligned}$$

(6.4)

(c) Let us estimate ${\tilde{a}}_{jk}(\lambda )$ for $j>k,\ \mu =\text {Re}\,\lambda \in [-2,0].$ By dividing the range of integration we obtain

$$\begin{aligned} \begin{aligned} {\tilde{a}}_{jk}(\lambda )&=\int _0^1 \frac{(1-s)^k}{(1+s)^{j+1}}\, s^{(j-k-\lambda )/2-1}\,\text {d}s+ \int _0^1 \frac{(1-s^{-1})^k}{(1+s^{-1})^{j+1}}\, s^{-(j-k-\lambda )/2-1}\,\text {d}s \\&=\int _0^1 \frac{(1-s)^k}{(1+s)^{j+1}}\, s^{(j-k)/2-1}\big [s^{-\lambda /2}+(-1)^ks^{\lambda /2+1}\big ]\,\text {d}s \end{aligned} \end{aligned}$$

and hence

$$\begin{aligned} |{\tilde{a}}_{jk}(\lambda )|\le 2 \int _0^1 \frac{(1-s)^k}{(1+s)^{j+1}}\, s^{(j-k)/2-1}\,\text {d}s. \end{aligned}$$

The standard integral representation [5, Equ. 9.111] of the hypergeometric function therefore yields

$$\begin{aligned} |{\tilde{a}}_{jk}(\lambda )|\le 2\,\frac{\Gamma \big (\frac{j-k}{2}\big )\,k!}{\Gamma (\frac{j+k}{2}+1)}\cdot {}_2F_1\big (j+1,\tfrac{j-k}{2};\tfrac{j+k}{2}+1;-1\big ). \end{aligned}$$

Upon inserting [2, Equ. 7.3.6.1] we obtain the following estimate of $|{\tilde{a}}_{jk}(\lambda )|$ by an upper bound independent of $\lambda$ and valid for $j>k,$$\mu =\text {Re}\,\lambda \in [-2,0]:$

$$\begin{aligned} |{\tilde{a}}_{jk}(\lambda )|\le 2^{-j}\Gamma \left( \frac{j-k}{2}\right) \,k!\sqrt{\pi }\Bigl [\frac{1}{\Gamma (\frac{j}{2}+1)\Gamma \big (\frac{k+1}{2}\big )}+ \frac{1}{\Gamma \big (\frac{j+1}{2}\big )\Gamma \big (\frac{k}{2}+1\big )}\Bigr ].\nonumber \\ \end{aligned}$$

(6.5)

(d) Combining formulas (6.4) and (6.5) and using the duplication formula for the gamma function yields, still for $j>k,$$\mu =\text {Re}\,\lambda \in [-2,0],$

$$\begin{aligned} |a_{jk}(\lambda )|\le \frac{\Gamma \big (\frac{j-k}{2}\big )}{|\Gamma \big (\frac{j-k-\lambda }{2}\big )|}\,\Bigl (\frac{\alpha _j}{\alpha _k}+\frac{\alpha _k}{\alpha _j}\Bigr ). \end{aligned}$$

Here $\alpha _j=\sqrt{\Gamma (\frac{j}{2}+1)/\Gamma (\frac{j+1}{2})}$ is defined as in Proposition 2. The limit [5, Equ. 8.328.2] implies

$$\begin{aligned} \exists C>0:\forall j,k\in {{\mathbb {N}}}_0\,\text { with }\,j\ge k: \frac{\Gamma (\frac{j-k}{2})}{|\Gamma \big (\frac{j-k-\lambda }{2}\big )|}\le C(1+j-k)^{\mu /2} \end{aligned}$$

with $\mu =\text {Re}\,\lambda .$ Hence the estimate of $\alpha _j/\alpha _k$ in formula (4.6) and the monotonicity of $\alpha _j$ imply

$$\begin{aligned} \exists C>0:\forall j,k\in {{\mathbb {N}}}_0\,\text { with }\,j> k:|a_{jk}(\lambda )|\le C(1+j-k)^{1/4+\mu /2}. \end{aligned}$$

Therefore, the estimate (6.1) holds in generality and this completes the proof. $\square$

Remark

Let us finally compare the assumption

$$\begin{aligned} F\in L^2({{\mathbb {S}}}^1),\text { i.e., }(c_k)_{k\in {{\mathbb {Z}}}}\in l^2({{\mathbb {Z}}}) \end{aligned}$$

(A)

with the hypothesis made in Proposition 4, namely

$$\begin{aligned} \bigl ((1+|k|)^{1/4+\mu /2}c_k\bigr )_{k\in {{\mathbb {Z}}}}\in l^1({{\mathbb {Z}}}). \end{aligned}$$

(B)

At the end of Sect. 2, we have seen that in the case of $-2<\mu =\text {Re}\, \lambda <-1,$ assumption (A) implies the boundedness of $W_\sigma$ on $L^2({{\mathbb {R}}})$ for $\sigma =F\cdot r^\lambda .$ Instead, assumption (B) implies that $W_\sigma$ is bounded whenever $\mu \in [-2,0]$ according to Proposition 4.

If $-2<\mu <-\frac{3}{2},$ then $\bigl ((1+|k|)^{1/4+\mu /2}\bigr )_{k\in {{\mathbb {Z}}}}\in l^2({{\mathbb {Z}}})$ and hence (A) implies (B). So condition (B) is the less restrictive and more general one. In contrast, when $-\frac{3}{2}\le \mu <-1,$ then neither of the two assumptions (A) and (B) implies the other one. In fact, e.g., for $\mu =-\frac{3}{2}$ and $c_k=(1+|k|)^{-1/2}/\log (1+|k|),$ condition (A) holds whereas (B) fails. Conversely, if $\mu =-\frac{3}{2}$ and $c_{k}=2^{j/2}j^{-2}$ for $k=2^j,$$j\in {{\mathbb {N}}},$ and $c_k=0$ for $k\not \in \{2^j;\,j\in {{\mathbb {N}}}\},$ then

$$\begin{aligned} \sum _{k\in {{\mathbb {Z}}}}(1+|k|)^{-1/2}|c_k|=\sum _{j=1}^\infty (1+2^j)^{-1/2}2^{j/2}j^{-2}\le \sum _{j=1}^\infty j^{-2}<\infty \end{aligned}$$

but $(c_k)_{k\in {{\mathbb {Z}}}}\not \in l^2({{\mathbb {Z}}}).$ 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 $W_\sigma$ if $\sigma$ 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 $W_\sigma .$ So the approach in Proposition 1 has its limitations from the word go.