Abstract
We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors O'C*(Rd) for tempered ultradistributions are analyzed via the duality with respect to the test function spaces OC*(Rd) introduced in this article. We also study ultradistribution spaces associated to translation-invariant Banach spaces of tempered ultradistributions and use their properties to provide a full characterization of the general convolution of Roumieu ultradistributions via the space of integrable ultradistributions. We show that the convolution of two Roumieu ultradistributions T,S∈D'{Mp}(Rd) exists if and only if (φSˇ)T∈D'L1{Mp}(Rd) for every φ∈D{Mp}(Rd).
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