Hamza Alzaareer, "Exponential Laws for Mixed-Order Ultradifferentiable Mappings," WSEAS Transactions on Mathematics, vol. 25, pp. 519-535, 2026, DOI:10.37394/23206.2026.25.48
Hamza Alzaareer. Exponential Laws for Mixed-Order Ultradifferentiable Mappings.
WSEAS Transactions on Mathematics. 2026;25:519-535. 10.37394/23206.2026.25.48
WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 25, 2026
Exponential Laws for Mixed-Order Ultradifferentiable Mappings
Author:
Abstract: We introduce the mixed-order classes $$C^{r,s}_{[\mathcal{M}]}$$ by combining mixed differentiability with the weight-matrix framework, in both the Beurling and Roumieu cases. At finite order we prove a chain rule under the Faà-di-Bruno property. The currying map is a linear injection. If $$r$$ and $$s$$ are finite, the class coincides with $$C^{r,s}$$ and the exponential law reduces to the mixed-order law without weights. Continuity of the inverse on the image holds when a lower weight bound is available at a single index.
Keywords:
Mixed-order differentiability, Ultradifferentiable mappings, Weight matrices, Exponential law, Faà-di-Bruno formula, Convenient calculus, Beurling and Roumieu classes, Chain rule
Pages: 519-535 WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 25, 2026, Art. #48 DOI: 10.37394/23206.2026.25.48
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