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

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.