Bulletin of the Belgian Mathematical Society - Simon Stevin

About spaces of $\omega_1$-$\omega_2$-ultradifferentiable functions

Jean Schmets and Manuel Valdivia

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Let $\Omega_1$ and $\Omega_2$ be non empty open subsets of $\mathbb R^r$ and $\mathbb R^s$ respectively and let $\omega_1$ and $\omega_2$ be weights. We introduce the spaces of ultradifferentiable functions $\mathcal{E}_{(\omega_1,\omega_2)}(\Omega_1 \times \Omega_2)$, $\mathcal{D}_{(\omega_1,\omega_2)}(\Omega_1 \times \Omega_2)$, $\mathcal{E}_{\{\omega_1,\omega_2\}}(\Omega_1 \times \Omega_2)$ and $\mathcal{D}_{\{\omega_1,\omega_2\}}(\Omega_1 \times \Omega_2)$, study their locally convex properties, examine the structure of their elements and also consider their links with the tensor products $\mathcal{E}_{*}(\Omega_1) \otimes \mathcal{E}_{*}(\Omega_2)$ and $\mathcal{D}_{*}(\Omega_1) \otimes \mathcal{D}_{*}(\Omega_2)$ endowed with the $\varepsilon$-, $\pi$- or $i$-topologies. This leads to kernel theorems.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 4 (2008), 645-662.

First available in Project Euclid: 5 November 2008

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Zentralblatt MATH identifier

Primary: 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A32: Spaces of linear operators; topological tensor products; approximation properties [See also 46B28, 46M05, 47L05, 47L20] 46E10: Topological linear spaces of continuous, differentiable or analytic functions 46F05: Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35]

ultradifferentiable functions Beurling type Roumieu type nuclearity tensor product kernel theorem


Schmets, Jean; Valdivia, Manuel. About spaces of $\omega_1$-$\omega_2$-ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 4, 645--662. doi:10.36045/bbms/1225893945. https://projecteuclid.org/euclid.bbms/1225893945

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