The authors introduce and study new classes of translation-invariant ultradistribution spaces which are natural generalizations of the spaces of ultradistributions with \(L^p\) growth.
The paper is organized as follows. Section 3 contains a characterization of tempered ultradistributions in terms of growth estimates for convolution averages. The convolutors for the space of tempered ultradistributions are described through duality with respect to a test function space constructed in this article. Section 4 is dedicated to the analysis of translation-invariant Banach spaces of tempered ultradistributions. It is proved that such a Banach space is a Banach module over the Beurling algebra \(L^1_\omega,\) were \(\omega(h)\) is the norm of the translation operator \(T_{-h}\) on the Banach space \(E.\) The classes of ultradifferentiable functions (of Beurling and Roumieu type) associated to a translation-invariant Banach space are introduced and studied in Section 5, while the corresponding classes of ultradistributions are discussed in Section 6. As important examples, the case that the Banach space \(E\) is a weighted \(L^p\) space is considered in Section 7.
The most important result of the present article is related to the convolution of Roumieu ultradistributions and is contained in the final Section 8. The authors show (Theorem 8.2) that the convolution of \(T,S\in {\mathcal D}^{'\left\{M_p\right\}}({\mathbb R}^d)\) exists if and only if \(\left(\varphi\ast\check{S}\right)T\in {\mathcal D}_{L^1}^{'\left\{M_p\right\}}({\mathbb R}^d)\) for every \(\varphi\in {\mathcal D}^{\left\{M_p\right\}}({\mathbb R}^d).\) This extends a previous result by R. Shiraishi on classical distributions [J. Sci. Hiroshima Univ., Ser. A 23, 19–32 (1959; Zbl 0091.28601)]. The existence of convolution for Beurling ultradistributions was already considered in the literature but the corresponding characterization in the Roumieu setting is more involved and has been a long-standing open question.