Kernel theorems in the setting of mixed nonquasi-analytic classes

Let Ω1 ⊂ Rr and Ω2 ⊂ Rs be nonempty and open. We introduce the Beurling–Roumieu spaces D(ω1,ω2}(Ω1 ×Ω2), D(M,M }(Ω1 ×Ω2) and obtain tensor product representations of them. This leads for instance to kernel theorems of the following type: every continuous linear map from the Beurling space D(ω1)(Ω1) (respectively D(M)(Ω1)) into the strong dual of the Roumieu space D{ω2}(Ω2) (respectively D{M }(Ω2)) can be represented by a continuous linear functional on D(ω1,ω2}(Ω1 ×Ω2) (respectively D(M,M }(Ω1 ×Ω2)).