MSC 26A33

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 …

Abstract. Let \(d\in\mathbb{N}\) and let \(\boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_d)\in(0,1)^d\). We prove that the algebraic direct sum \[ \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} := \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C}e_{\mathbf{k}}, \quad e_{\mathbf{k}}(x) := \prod_{j=1}^d \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}, \] is the canonical multi-graded monomial space on which the partial Riemann–Liouville integrals \( J_j:={}_0 I_{x_j}^{\alpha_j} \text{ for } 1\le j\le d \) and the partial Caputo derivatives …