A Spectral Algebra for Multidimensional Weyl Fractional Operators on Exponential Characters

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 $W^{{\alpha}}$ of order ${\alpha}\in[0,\infty)^d$, we obtain$$ W^{{\alpha}}e_\lambda=\lambda^{{\alpha}}e_\lambda, \qquad \lambda^{{\alpha}}:=\prod_{j=1}^d \lambda_j^{\alpha_j},$$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.