Abstract. Let \(d\in\mathbb{N}\). We show that multidimensional Weyl fractional operators admit a natural spectral realization on the algebraic span of exponential characters. More precisely, if \(\Lambda\subset (\mathbb{C}_+)^d\) and \[ e_\lambda(x):=e^{\langle \lambda,x\rangle}, \qquad \lambda\in\Lambda,\ x\in\mathbb{R}^d, \] where \(\mathbb{C}_+:=\{z\in\mathbb{C}:\Re z>0\}\), then the algebraic direct sum \[ \mathcal{E}_{\Lambda}^{\mathrm{alg}} := \bigoplus_{\lambda\in\Lambda}\mathbb{C}e_\lambda \] carries a canonical diagonal action of the generalized multidimensional Weyl integrals and derivatives. If \[ I_{W,a}=I_1^{a_1}\cdots I_d^{a_d} \qquad\text{and}\qquad D_{W,a,m}=D_1^{a_1,m_1}\cdots D_d^{a_d,m_d} \] are the coordinatewise generalized Weyl integral and derivative determined by a kernel tuple \(a=(a_1,\dots,a_d)\) and an integer multi-index \(m\in\mathbb{N}_0^d\), then \[ I_{W,a}e_\lambda=\widehat{a}(\lambda)e_\lambda \qquad\text{and}\qquad D_{W,a,m}e_\lambda=\sigma_{a,m}(\lambda)e_\lambda, \] with symbols \[ \widehat{a}(\lambda):=\prod_{j=1}^d \widehat{a_j}(\lambda_j), \qquad \sigma_{a,m}(\lambda):=\prod_{j=1}^d \lambda_j^{m_j}\widehat{a_j}(\lambda_j). \] Thus the corresponding operator algebra is diagonal and isomorphic to the algebra of scalar multipliers on \(\Lambda\). We further prove the law of exponents by symbol multiplication: \[ I_{W,a}I_{W,b}=I_{W,a*_0 b}, \qquad D_{W,a,m}D_{W,b,n}=D_{W,a*_0 b,m+n}. \] For the standard Weyl fractional derivative \(W^{\boldsymbol{\alpha}}\) of order \(\boldsymbol{\alpha}\in[0,\infty)^d\), we obtain \[ W^{\boldsymbol{\alpha}}e_\lambda=\lambda^{\boldsymbol{\alpha}}e_\lambda, \qquad \lambda^{\boldsymbol{\alpha}}:=\prod_{j=1}^d \lambda_j^{\alpha_j}, \] so the Weyl calculus becomes a genuine spectral algebra on \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\). This stands in contrast to the shift-algebra models for Riemann–Liouville and Caputo operators on half-spaces: on \(\mathbb{R}^d\) there is no vacuum defect and no boundary layer.
Keywords. multidimensional Weyl fractional calculus; generalized Weyl derivative; spectral algebra; exponential characters; law of exponents; symbol calculus.
MSC 2020. 26A33; 47A60; 35S05.
1. Introduction
Fractional calculus may be organized in several distinct operator-theoretic ways, depending on the ambient domain and on the type of fractional operator under consideration; see, for example, [4, 5, 6, 7]. In particular, the Riemann–Liouville and Caputo operators on a half-line or a half-space are naturally tied to boundary data and initial traces, whereas Weyl-type operators on the whole real line or on \(\mathbb{R}^d\) are naturally tied to translation-invariant structures.
This distinction becomes especially transparent in algebraic models. In [2], it was shown that on the canonical \(\alpha\)-graded monomial chain \[ \mathcal{G}_{\alpha}^{\mathrm{alg}} = \bigoplus_{n=0}^{\infty}\mathbb{C}\frac{x^{n\alpha}}{\Gamma(n\alpha+1)}, \] the order-\(\alpha\) Riemann–Liouville integral and the order-\(\alpha\) Caputo derivative form a unilateral shift pair. In [3], this was extended to the multidimensional setting, where partial Riemann–Liouville integrals and partial Caputo derivatives realize a commuting shift algebra on a canonical multi-graded monomial lattice. The finite-dimensional obstruction to similar models on ordinary polynomial spaces was analyzed in [1].
The purpose of the present paper is to develop the corresponding whole-space picture for Weyl fractional operators. Here the shift-algebra viewpoint is no longer the natural one. Because the Weyl calculus is translation-invariant and possesses a clean law of exponents on appropriate function classes, the correct algebraic model is spectral rather than shift-theoretic. The canonical basis is no longer monomial but exponential.
This perspective is already suggested by Kostić's multidimensional Weyl framework [8]. In particular, Kostić shows that generalized multidimensional Weyl operators admit a law of exponents and that exponential functions diagonalize the coordinatewise Weyl derivatives. The present paper extracts from that setting a precise algebraic consequence: on the algebraic span of exponential characters, the multidimensional Weyl calculus becomes a diagonal multiplier algebra.
More concretely, let \[ \mathbb{C}_+:=\{z\in\mathbb{C}:\Re z>0\}, \] and let \(\Lambda\subset (\mathbb{C}_+)^d\). For each \(\lambda=(\lambda_1,\dots,\lambda_d)\in\Lambda\), define the exponential character \[ e_\lambda(x):=e^{\langle \lambda,x\rangle}, \qquad \langle \lambda,x\rangle:=\lambda_1x_1+\cdots+\lambda_d x_d. \] We consider the algebraic direct sum \[ \mathcal{E}_{\Lambda}^{\mathrm{alg}} := \bigoplus_{\lambda\in\Lambda}\mathbb{C}e_\lambda. \] If \(a=(a_1,\dots,a_d)\) is a tuple of one-sided kernels and \(m=(m_1,\dots,m_d)\in\mathbb{N}_0^d\), then the coordinatewise generalized Weyl integral and derivative act on the basis by scalar multiplication: \[ I_{W,a}e_\lambda=\widehat{a}(\lambda)e_\lambda, \qquad D_{W,a,m}e_\lambda=\sigma_{a,m}(\lambda)e_\lambda. \] It follows that the entire algebra generated by these operators is diagonal and can be identified with a symbol algebra on \(\Lambda\).
The main contributions of this paper are as follows.
- We construct the algebraic spectral module \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\) and prove that generalized multidimensional Weyl operators act diagonally on its basis.
- We identify the resulting operator algebra with the full algebra of scalar multipliers on \(\Lambda\).
- We derive the law of exponents entirely from symbol multiplication.
- We recover the standard Weyl fractional derivatives as the special case whose symbols are \(\lambda^{\boldsymbol{\alpha}}\).
- We apply the spectral algebra to constant-coefficient Weyl equations and explain the contrast with the shift-algebra models for Riemann–Liouville and Caputo operators.
Throughout the paper, all vectors are complex-valued, all sums in \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\) are finite, and all identities are understood pointwise on \(\mathbb{R}^d\). For each real \(\gamma\ge 0\) and \(z\in\mathbb{C}_+\), we use the principal branch \[ z^\gamma:=e^{\gamma\operatorname{Log} z}, \] where \(\operatorname{Log}\) denotes the principal logarithm on \(\mathbb{C}_+\).
2. Exponential characters and admissible Weyl kernels
We begin by recalling the exponential basis and the kernel class that will be used throughout the paper.
Definition 2.1. Let \(\Lambda\subset (\mathbb{C}_+)^d\). For each \(\lambda\in\Lambda\), define \[ e_\lambda(x):=e^{\langle \lambda,x\rangle}, \qquad x\in\mathbb{R}^d. \] The algebraic spectral module over \(\Lambda\) is the algebraic direct sum \[ \mathcal{E}_{\Lambda}^{\mathrm{alg}} := \bigoplus_{\lambda\in\Lambda}\mathbb{C}e_\lambda. \]
Remark 2.2. The family \(\{e_\lambda\}_{\lambda\in\Lambda}\) is linearly independent. Indeed, if \[ \sum_{\nu=1}^N c_\nu e_{\lambda^{(\nu)}}(x)=0 \qquad (x\in\mathbb{R}^d) \] for distinct \(\lambda^{(1)},\dots,\lambda^{(N)}\in\Lambda\), then restricting to the line \[ x=t\xi \] with a generic \(\xi\in\mathbb{R}^d\) reduces the identity to a finite linear combination of distinct one-variable exponentials, which is possible only when all coefficients vanish.
Proposition 2.3. Let \(\lambda=(\lambda_1,\dots,\lambda_d)\in\mathbb{C}^d\), and let \(f\in C^1(\mathbb{R}^d)\) satisfy \[ \partial_j f=\lambda_j f \qquad (1\le j\le d). \] Then there exists a constant \(c\in\mathbb{C}\) such that \[ f(x)=c\,e^{\langle \lambda,x\rangle} \qquad (x\in\mathbb{R}^d). \]
Proof.
Define \[ g(x):=e^{-\langle \lambda,x\rangle}f(x). \] Then, for each \(j\), \[ \partial_j g = -\lambda_j e^{-\langle \lambda,x\rangle}f(x) + e^{-\langle \lambda,x\rangle}\partial_j f(x) = -\lambda_j g(x)+\lambda_j g(x) = 0. \] Hence all first partial derivatives of \(g\) vanish, so \(g\) is constant on \(\mathbb{R}^d\). Therefore \[ f(x)=c\,e^{\langle \lambda,x\rangle} \] for some \(c\in\mathbb{C}\).
Remark 2.4. Proposition 2.3 shows that exponential characters are the canonical joint eigenfunctions of the ordinary derivative tuple \[ (\partial_1,\dots,\partial_d). \] This is the spectral analogue of the canonical monomial basis that appears in the shift-algebra models for Riemann–Liouville and Caputo operators [2, 3].
Definition 2.5. A tuple of kernels \[ a=(a_1,\dots,a_d) \] is called admissible if each \[ a_j\in L^1_{\mathrm{loc}}([0,\infty)) \] and its Laplace transform \[ \widehat{a_j}(z) := \int_0^\infty a_j(s)e^{-zs}\,ds \] exists for every \(z\in\mathbb{C}_+\).
If \(a=(a_1,\dots,a_d)\) and \(b=(b_1,\dots,b_d)\) are admissible, we define their coordinatewise one-sided convolution by \[ a*_0 b := (a_1*_0 b_1,\dots,a_d*_0 b_d), \] where \[ (a_j*_0 b_j)(t):=\int_0^t a_j(t-s)b_j(s)\,ds \qquad (t\ge 0). \]
Remark 2.6. For admissible \(a\) and \(\lambda=(\lambda_1,\dots,\lambda_d)\in(\mathbb{C}_+)^d\), we write \[ \widehat{a}(\lambda) := \prod_{j=1}^d \widehat{a_j}(\lambda_j). \] By the usual Laplace-transform identity, \[ \widehat{a_j*_0 b_j}(z)=\widehat{a_j}(z)\widehat{b_j}(z) \qquad (z\in\mathbb{C}_+), \] and therefore \[ \widehat{a*_0 b}(\lambda)=\widehat{a}(\lambda)\widehat{b}(\lambda) \qquad (\lambda\in(\mathbb{C}_+)^d). \]
3. Generalized multidimensional Weyl operators on the spectral module
We now define the generalized Weyl operators on \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\) and compute their action on the exponential basis.
Let \(\varepsilon_j\in\mathbb{R}^d\) denote the \(j\)-th standard basis vector.
Definition 3.1. Let \(a=(a_1,\dots,a_d)\) be an admissible kernel tuple, and let \[ m=(m_1,\dots,m_d)\in\mathbb{N}_0^d. \] For each \(j\in\{1,\dots,d\}\) and each \(u\in\mathcal{E}_{\Lambda}^{\mathrm{alg}}\), define \[ (I_j^{a_j}u)(x) := \int_0^\infty a_j(s)u(x-s\varepsilon_j)\,ds, \] and \[ D_j^{a_j,m_j}u := \partial_j^{m_j}(I_j^{a_j}u). \] We then define the multidimensional generalized Weyl integral and derivative by \[ I_{W,a}:=I_1^{a_1}\cdots I_d^{a_d}, \] and \[ D_{W,a,m}:=D_1^{a_1,m_1}\cdots D_d^{a_d,m_d}. \]
Remark 3.2. Definition 3.1 agrees, on the exponential spectral module, with the coordinatewise generalized Weyl calculus considered by Kostić [8]. Since every element of \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\) is a finite linear combination of exponentials and each \(\lambda_j\) lies in \(\mathbb{C}_+\), all defining integrals converge absolutely on basis vectors and therefore on all of \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\).
Lemma 3.3. Let \(a=(a_1,\dots,a_d)\) be admissible, let \(m\in\mathbb{N}_0^d\), let \(\lambda\in\Lambda\), and let \(j\in\{1,\dots,d\}\). Then \[ I_j^{a_j}e_\lambda = \widehat{a_j}(\lambda_j)e_\lambda, \] and \[ D_j^{a_j,m_j}e_\lambda = \lambda_j^{m_j}\widehat{a_j}(\lambda_j)e_\lambda. \]
Proof.
By definition, \[ (I_j^{a_j}e_\lambda)(x) = \int_0^\infty a_j(s)e^{\langle \lambda,x-s\varepsilon_j\rangle}\,ds = e^{\langle \lambda,x\rangle} \int_0^\infty a_j(s)e^{-\lambda_j s}\,ds = \widehat{a_j}(\lambda_j)e_\lambda(x). \] Applying \(\partial_j^{m_j}\) yields \[ D_j^{a_j,m_j}e_\lambda = \partial_j^{m_j}\!\left(\widehat{a_j}(\lambda_j)e_\lambda\right) = \widehat{a_j}(\lambda_j)\lambda_j^{m_j}e_\lambda.\tag*{\(\square\)} \]
Theorem 3.4. Let \(a=(a_1,\dots,a_d)\) be admissible, and let \[ m=(m_1,\dots,m_d)\in\mathbb{N}_0^d. \] For every \(\lambda\in\Lambda\), \[ I_{W,a}e_\lambda=\widehat{a}(\lambda)e_\lambda, \] and \[ D_{W,a,m}e_\lambda=\sigma_{a,m}(\lambda)e_\lambda, \] where \[ \widehat{a}(\lambda):=\prod_{j=1}^d \widehat{a_j}(\lambda_j), \qquad \sigma_{a,m}(\lambda):=\prod_{j=1}^d \lambda_j^{m_j}\widehat{a_j}(\lambda_j). \]
Proof.
Since the coordinate operators act diagonally on each basis vector by Lemma 3.3, one has \[ I_{W,a}e_\lambda = I_1^{a_1}\cdots I_d^{a_d}e_\lambda = \prod_{j=1}^d \widehat{a_j}(\lambda_j)e_\lambda = \widehat{a}(\lambda)e_\lambda. \] The proof for \(D_{W,a,m}\) is identical: \[ D_{W,a,m}e_\lambda = D_1^{a_1,m_1}\cdots D_d^{a_d,m_d}e_\lambda = \prod_{j=1}^d \lambda_j^{m_j}\widehat{a_j}(\lambda_j)e_\lambda = \sigma_{a,m}(\lambda)e_\lambda.\tag*{\(\square\)} \]
Corollary 3.5. For all admissible kernel tuples \(a,b\) and all integer multi-indices \(m,n\in\mathbb{N}_0^d\), the operators \[ I_{W,a},\ I_{W,b},\ D_{W,a,m},\ D_{W,b,n} \] pairwise commute on \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\).
Proof.
By Theorem 3.4, each of the above operators acts diagonally on the basis \(\{e_\lambda\}_{\lambda\in\Lambda}\). Diagonal operators in a fixed basis commute.
4. The spectral multiplier algebra
The diagonal form of Theorem 3.4 leads naturally to a multiplier algebra.
Definition 4.1. Let \(\Lambda\subset(\mathbb{C}_+)^d\), and let \(\sigma:\Lambda\to\mathbb{C}\) be any function. Define the corresponding multiplier \[ M_\sigma:\mathcal{E}_{\Lambda}^{\mathrm{alg}} \longrightarrow \mathcal{E}_{\Lambda}^{\mathrm{alg}} \] by \[ M_\sigma\!\left(\sum_{\lambda\in F} c_\lambda e_\lambda\right) := \sum_{\lambda\in F} c_\lambda \sigma(\lambda)e_\lambda, \] where \(F\subset\Lambda\) is finite.
Theorem 4.2. Let \[ \mathscr{M}(\Lambda):=\{\,M_\sigma:\sigma:\Lambda\to\mathbb{C}\,\}. \] Then:
- \(\mathscr{M}(\Lambda)\) is a commutative subalgebra of \(\operatorname{End}(\mathcal{E}_{\Lambda}^{\mathrm{alg}})\).
- The map \[ \sigma\longmapsto M_\sigma \] is an algebra isomorphism from the algebra of all scalar functions on \(\Lambda\) (with pointwise operations) onto \(\mathscr{M}(\Lambda)\).
- Every operator in \(\mathscr{M}(\Lambda)\) is diagonal in the exponential basis \(\{e_\lambda\}_{\lambda\in\Lambda}\), and every diagonal operator in that basis belongs to \(\mathscr{M}(\Lambda)\).
Proof.
For any functions \(\sigma,\tau:\Lambda\to\mathbb{C}\) and any basis vector \(e_\lambda\), \[ M_\sigma M_\tau e_\lambda = M_\sigma(\tau(\lambda)e_\lambda) = \sigma(\lambda)\tau(\lambda)e_\lambda = M_{\sigma\tau}e_\lambda. \] Hence \[ M_\sigma M_\tau = M_{\sigma\tau}, \] so \(\mathscr{M}(\Lambda)\) is a commutative algebra and the map \(\sigma\mapsto M_\sigma\) preserves multiplication. It obviously preserves addition and scalar multiplication as well.
The map is injective because \[ M_\sigma e_\lambda = \sigma(\lambda)e_\lambda \] for each \(\lambda\in\Lambda\). It is surjective by definition of \(\mathscr{M}(\Lambda)\). This proves (i) and (ii).
For (iii), every multiplier is diagonal by construction. Conversely, if \(T\) is diagonal in the basis \(\{e_\lambda\}_{\lambda\in\Lambda}\), define \[ \sigma(\lambda)\in\mathbb{C} \quad\text{by}\quad Te_\lambda=\sigma(\lambda)e_\lambda. \] Then \(T=M_\sigma\).
Corollary 4.3. Let \(a=(a_1,\dots,a_d)\) be admissible and let \(m\in\mathbb{N}_0^d\). Then \[ I_{W,a}=M_{\widehat{a}}, \qquad D_{W,a,m}=M_{\sigma_{a,m}}, \] where \[ \widehat{a}(\lambda)=\prod_{j=1}^d \widehat{a_j}(\lambda_j), \qquad \sigma_{a,m}(\lambda)=\prod_{j=1}^d \lambda_j^{m_j}\widehat{a_j}(\lambda_j). \]
Proof.
This is exactly the content of Theorem 3.4, rephrased in the language of Definition 4.1.
Theorem 4.4 (Generalized law of exponents). Let \(a,b\) be admissible kernel tuples, and let \(m,n\in\mathbb{N}_0^d\). Then \[ I_{W,a}I_{W,b}=I_{W,a*_0 b}, \] and \[ D_{W,a,m}D_{W,b,n}=D_{W,a*_0 b,m+n}. \]
Proof.
By Corollary 4.3, \[ I_{W,a}I_{W,b} = M_{\widehat{a}}M_{\widehat{b}} = M_{\widehat{a}\,\widehat{b}}. \] Since \[ \widehat{a*_0 b}(\lambda) = \prod_{j=1}^d \widehat{a_j*_0 b_j}(\lambda_j) = \prod_{j=1}^d \widehat{a_j}(\lambda_j)\widehat{b_j}(\lambda_j) = \widehat{a}(\lambda)\widehat{b}(\lambda), \] we obtain \[ I_{W,a}I_{W,b} = M_{\widehat{a*_0 b}} = I_{W,a*_0 b}. \]
Likewise, \[ D_{W,a,m}D_{W,b,n} = M_{\sigma_{a,m}}M_{\sigma_{b,n}} = M_{\sigma_{a,m}\sigma_{b,n}}. \] Now \[ \sigma_{a,m}(\lambda)\sigma_{b,n}(\lambda) = \prod_{j=1}^d \lambda_j^{m_j+n_j}\widehat{a_j}(\lambda_j)\widehat{b_j}(\lambda_j) = \prod_{j=1}^d \lambda_j^{m_j+n_j}\widehat{a_j*_0 b_j}(\lambda_j) = \sigma_{a*_0 b,m+n}(\lambda). \] Hence \[ D_{W,a,m}D_{W,b,n} = M_{\sigma_{a*_0 b,m+n}} = D_{W,a*_0 b,m+n}.\tag*{\(\square\)} \]
Remark 4.5. Theorem 4.4 is the algebraic spectral form of the law of exponents for generalized multidimensional Weyl operators. On the spectral module, the proof is reduced entirely to the multiplication of symbols.
5. Standard multidimensional Weyl fractional derivatives
We now recover the standard Weyl fractional derivatives from the generalized kernel calculus.
Lemma 5.1. Let \(\nu>0\), and define \[ g_\nu(s):=\frac{s^{\nu-1}}{\Gamma(\nu)} \qquad (s>0). \] Then for every \(z\in\mathbb{C}_+\), \[ \int_0^\infty g_\nu(s)e^{-zs}\,ds=z^{-\nu}, \] where the principal branch of the complex power is used.
Proof.
Since \(\Re z>0\), the substitution \(u=zs\) yields \[ \int_0^\infty \frac{s^{\nu-1}}{\Gamma(\nu)}e^{-zs}\,ds = \frac{1}{\Gamma(\nu)}z^{-\nu}\int_0^\infty u^{\nu-1}e^{-u}\,du = \frac{\Gamma(\nu)}{\Gamma(\nu)}z^{-\nu} = z^{-\nu}.\tag*{\(\square\)} \]
Definition 5.2. Let \(\beta\ge 0\) and \(j\in\{1,\dots,d\}\).
- If \(\beta=0\), define \[ W_j^0:=I. \]
- If \(\beta\in\mathbb{N}\), define \[ W_j^\beta:=\partial_j^\beta. \]
- If \(\beta\notin\mathbb{N}\) and \(m:=\lceil \beta\rceil\), define \[ W_j^\beta:=D_j^{g_{m-\beta},m}. \]
For \[ \boldsymbol{\beta}=(\beta_1,\dots,\beta_d)\in[0,\infty)^d, \] define \[ W^{\boldsymbol{\beta}} := W_1^{\beta_1}\cdots W_d^{\beta_d}. \] This is well defined on \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\) by Corollary 3.5.
Theorem 5.3. Let \[ \boldsymbol{\beta}=(\beta_1,\dots,\beta_d)\in[0,\infty)^d. \] Then, for every \(\lambda=(\lambda_1,\dots,\lambda_d)\in\Lambda\), \[ W_j^{\beta_j}e_\lambda=\lambda_j^{\beta_j}e_\lambda \qquad (1\le j\le d), \] and \[ W^{\boldsymbol{\beta}}e_\lambda = \lambda^{\boldsymbol{\beta}}e_\lambda, \qquad \lambda^{\boldsymbol{\beta}} := \prod_{j=1}^d \lambda_j^{\beta_j}. \]
Proof.
Fix \(j\in\{1,\dots,d\}\). If \(\beta_j=0\), then \[ W_j^{\beta_j}e_\lambda=Ie_\lambda=e_\lambda=\lambda_j^0e_\lambda. \] If \(\beta_j\in\mathbb{N}\), then \[ W_j^{\beta_j}e_\lambda = \partial_j^{\beta_j}e_\lambda = \lambda_j^{\beta_j}e_\lambda. \] Suppose now that \(\beta_j\notin\mathbb{N}\), and write \(m_j:=\lceil \beta_j\rceil\). Then \[ W_j^{\beta_j} = D_j^{g_{m_j-\beta_j},m_j}. \] By Lemma 5.1, \[ \widehat{g_{m_j-\beta_j}}(\lambda_j)=\lambda_j^{-(m_j-\beta_j)}. \] Therefore Lemma 3.3 yields \[ W_j^{\beta_j}e_\lambda = \lambda_j^{m_j}\widehat{g_{m_j-\beta_j}}(\lambda_j)e_\lambda = \lambda_j^{m_j}\lambda_j^{-(m_j-\beta_j)}e_\lambda = \lambda_j^{\beta_j}e_\lambda. \]
Since the coordinate operators commute and each acts by scalar multiplication on \(e_\lambda\), we obtain \[ W^{\boldsymbol{\beta}}e_\lambda = \prod_{j=1}^d \lambda_j^{\beta_j}e_\lambda = \lambda^{\boldsymbol{\beta}}e_\lambda.\tag*{\(\square\)} \]
Corollary 5.4 (Law of exponents for standard Weyl derivatives). Let \[ \boldsymbol{\alpha},\boldsymbol{\beta}\in[0,\infty)^d. \] Then \[ W^{\boldsymbol{\alpha}}W^{\boldsymbol{\beta}} = W^{\boldsymbol{\alpha}+\boldsymbol{\beta}} = W^{\boldsymbol{\beta}}W^{\boldsymbol{\alpha}} \] on all of \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\).
Proof.
It suffices to verify the identity on basis vectors. By Theorem 5.3, \[ W^{\boldsymbol{\alpha}}W^{\boldsymbol{\beta}}e_\lambda = \lambda^{\boldsymbol{\alpha}}\lambda^{\boldsymbol{\beta}}e_\lambda. \] Because each \(\lambda_j\in\mathbb{C}_+\) and we use the principal branch on \(\mathbb{C}_+\), one has \[ \lambda_j^{\alpha_j}\lambda_j^{\beta_j} = \lambda_j^{\alpha_j+\beta_j}. \] Therefore \[ \lambda^{\boldsymbol{\alpha}}\lambda^{\boldsymbol{\beta}} = \lambda^{\boldsymbol{\alpha}+\boldsymbol{\beta}}, \] and hence \[ W^{\boldsymbol{\alpha}}W^{\boldsymbol{\beta}}e_\lambda = \lambda^{\boldsymbol{\alpha}+\boldsymbol{\beta}}e_\lambda = W^{\boldsymbol{\alpha}+\boldsymbol{\beta}}e_\lambda. \] The symmetry in \(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\) is immediate.
Remark 5.5. Corollary 5.4 shows that, on the spectral module, the standard multidimensional Weyl calculus is a genuine abelian functional calculus. In particular, there are no defect projections analogous to those that appear in the Caputo shift algebra on a half-space [3].
6. Symbolic solvability of constant-coefficient Weyl equations
The diagonal spectral form makes constant-coefficient equations completely transparent.
Theorem 6.1. Let \(\sigma:\Lambda\to\mathbb{C}\). Then the multiplier \(M_\sigma\) is invertible on \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\) if and only if \[ \sigma(\lambda)\neq 0 \qquad (\lambda\in\Lambda). \] In that case, \[ M_\sigma^{-1}=M_{\sigma^{-1}}, \qquad \sigma^{-1}(\lambda):=\frac{1}{\sigma(\lambda)}. \]
Proof.
If \(M_\sigma\) is invertible, then \[ 0\neq e_\lambda = M_\sigma^{-1}M_\sigma e_\lambda = \sigma(\lambda)M_\sigma^{-1}e_\lambda, \] so \(\sigma(\lambda)\neq 0\) for every \(\lambda\in\Lambda\).
Conversely, if \(\sigma(\lambda)\neq 0\) for all \(\lambda\), then \[ M_{\sigma^{-1}}M_\sigma e_\lambda = \sigma^{-1}(\lambda)\sigma(\lambda)e_\lambda = e_\lambda, \] and similarly \[ M_\sigma M_{\sigma^{-1}}e_\lambda=e_\lambda. \] Thus \[ M_\sigma^{-1}=M_{\sigma^{-1}}.\tag*{\(\square\)} \]
Definition 6.2. Let \[ \boldsymbol{\beta}=(\beta_1,\dots,\beta_d)\in[0,\infty)^d, \] and let \[ p(z_1,\dots,z_d)=\sum_{\nu\in F} c_\nu z^\nu \] be a complex polynomial in \(d\) variables, where \(F\subset\mathbb{N}_0^d\) is finite and \[ z^\nu:=z_1^{\nu_1}\cdots z_d^{\nu_d}. \] We define the corresponding constant-coefficient Weyl operator on \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\) by \[ p(W^{\boldsymbol{\beta}}) := \sum_{\nu\in F} c_\nu \prod_{j=1}^d \big(W_j^{\beta_j}\big)^{\nu_j}. \]
Theorem 6.3. Let \(p\) and \(\boldsymbol{\beta}\) be as in Definition 6.2. Then, for every \(\lambda\in\Lambda\), \[ p(W^{\boldsymbol{\beta}})e_\lambda = p\big(\lambda_1^{\beta_1},\dots,\lambda_d^{\beta_d}\big)e_\lambda. \] Equivalently, \[ p(W^{\boldsymbol{\beta}}) = M_{\rho_p}, \] where \[ \rho_p(\lambda) := p\big(\lambda_1^{\beta_1},\dots,\lambda_d^{\beta_d}\big). \]
Proof.
By Corollary 5.4, \[ \prod_{j=1}^d \big(W_j^{\beta_j}\big)^{\nu_j} = W^{\nu\odot\boldsymbol{\beta}}. \] Hence Theorem 5.3 gives \[ \left(\prod_{j=1}^d \big(W_j^{\beta_j}\big)^{\nu_j}\right)e_\lambda = \prod_{j=1}^d \lambda_j^{\nu_j\beta_j}e_\lambda = \big(\lambda_1^{\beta_1}\big)^{\nu_1}\cdots \big(\lambda_d^{\beta_d}\big)^{\nu_d}e_\lambda. \] Summing over \(\nu\in F\), we obtain \[ p(W^{\boldsymbol{\beta}})e_\lambda = \sum_{\nu\in F} c_\nu \big(\lambda_1^{\beta_1}\big)^{\nu_1}\cdots \big(\lambda_d^{\beta_d}\big)^{\nu_d}e_\lambda = p\big(\lambda_1^{\beta_1},\dots,\lambda_d^{\beta_d}\big)e_\lambda. \] This is precisely the multiplier identity.
Corollary 6.4. Let \(p\) and \(\boldsymbol{\beta}\) be as in Definition 6.2. The equation \[ p(W^{\boldsymbol{\beta}})u=f \] has a unique solution \[ u\in\mathcal{E}_{\Lambda}^{\mathrm{alg}} \] for every \[ f\in\mathcal{E}_{\Lambda}^{\mathrm{alg}} \] if and only if \[ p\big(\lambda_1^{\beta_1},\dots,\lambda_d^{\beta_d}\big)\neq 0 \qquad (\lambda\in\Lambda). \] In that case, \[ u = M_{\rho_p^{-1}}f, \qquad \rho_p^{-1}(\lambda) := \frac{1}{p(\lambda_1^{\beta_1},\dots,\lambda_d^{\beta_d})}. \]
Proof.
By Theorem 6.3, \[ p(W^{\boldsymbol{\beta}})=M_{\rho_p}. \] The result therefore follows immediately from Theorem 6.1.
Remark 6.5. Corollary 6.4 shows that on the algebraic spectral module, constant-coefficient multidimensional Weyl equations reduce completely to pointwise spectral division. In this sense, the Weyl calculus behaves here exactly like a diagonal functional calculus.
7. Comparison with shift-algebra models
Remark 7.1. The shift-algebra model of [3] is built on the multi-graded monomial lattice \[ \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C} \prod_{j=1}^d\frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)} \] over the domain \([0,\infty)^d\). There, the partial Riemann–Liouville integrals act as forward shifts and the partial Caputo derivatives act as backward shifts, with coordinate-vacuum annihilation and higher-order boundary defects.
By contrast, the present Weyl model is built on the spectral module \[ \mathcal{E}_{\Lambda}^{\mathrm{alg}} = \bigoplus_{\lambda\in\Lambda}\mathbb{C}e^{\langle \lambda,x\rangle} \] over the whole space \(\mathbb{R}^d\). Here the operators are diagonal rather than shift-like, and there is no vacuum sector, no boundary layer, and no defect projection.
Remark 7.2. This distinction is not superficial. It reflects the difference between one-sided fractional operators, which are anchored at a boundary and therefore carry initial-data defects, and Weyl-type operators on the whole space, which are translation-invariant and therefore admit a clean spectral realization. In this sense, the shift-algebra and spectral-algebra models are complementary rather than competing descriptions.
Remark 7.3. The present paper remains algebraic in spirit. No completeness or topology is imposed on \(\mathcal{E}_{\Lambda}^{\mathrm{alg}}\), and all sums are finite. A natural next step would be to pass to Fourier–Laplace or Banach-space completions in which the diagonal multiplier algebra becomes an analytic functional calculus for multidimensional Weyl operators. This lies closer to the broader program initiated in [8].
8. Conclusion
We have shown that multidimensional generalized Weyl fractional operators admit a natural spectral realization on the algebraic span of exponential characters. The central object is the spectral module \[ \mathcal{E}_{\Lambda}^{\mathrm{alg}} = \bigoplus_{\lambda\in\Lambda}\mathbb{C}e_\lambda, \qquad e_\lambda(x)=e^{\langle \lambda,x\rangle}, \] where \[ \Lambda\subset(\mathbb{C}_+)^d. \] On this module, generalized multidimensional Weyl integrals and derivatives act diagonally, and the operator algebra they generate is identified with a multiplier algebra on the spectral set \(\Lambda\).
This gives a concrete algebraic form to several structural features of the multidimensional Weyl calculus:
- generalized Weyl operators are diagonal in the exponential basis;
- the law of exponents is simply symbol multiplication;
- the standard Weyl fractional derivatives have the exact symbols \(\lambda^{\boldsymbol{\alpha}}\);
- constant-coefficient Weyl equations reduce to pointwise algebra on the spectrum.
From this perspective, the multidimensional Weyl calculus is naturally a spectral algebra. This stands in deliberate contrast to the shift-algebra models for Riemann–Liouville and Caputo operators on half-spaces [2, 3], where one-sidedness and boundary data force vacuum and defect structures. The two pictures are therefore complementary manifestations of fractional calculus on different geometric domains.
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