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

Abstract

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.