Convolution and Localization Operators in Ultradistribution Spaces

Abstract

We investigate the Laplace transform in Komatsu ultradistributions and give conditions under which an analytic function is a Laplace transformation of an ultradistribution. We prove the equivalence of several denitions of convolu-tion of two Roumieu ultradistributions. For that purpose, we consider the _ ten-sor product of _~BfMpg and a locally convex space. We dene specic global symbol classes of Shubin type and study the corresponding pseudodierential operators of innite order that act continuously on the spaces of tempered ultradistributions of Beurling and Roumieu type. For these classes, we develop symbolic calculus. We investigate the connection between the Anti-Wick and Weyl quantization when the symbols belong to these classes. We nd the largest subspace of ultradistri-butions for which the convolution with the gaussian kernel exists. This gives a way to extend the denition of Anti-Wick quantization for symbols that are not necessarily tempered ultradistributions.

Prouqavamo Laplasovu transformaciju u prostorima Komat-suove ultradistribucije i dajemo uslov pod kojim analitiqka funk-cija je Laplasova transformacija ultradistribucije. Dokazujemo ek-vivalentnost nekoliko definicija o konvoluciji dve Rumie ultradis-tribucije. Za ovu svrhu razmatramo " tenzorski proizvod ~ B fMpg i lokalno konveksni prostor. Definiramo specifiqne globalne simbol klase Xubinovog tipa i prouqavamo odgovarajue psevdo diferenci-jalne operatore beskonaqnog reda koji neprekidno deluju na prosto-rima temperiranih ultradistribucija Berlineovog i Rumieovog tipa. Za ove klase gradimo simboliqki kalkulus. Prouqavamo vezu izmeu Anti-Wick-ove i Weyl-ove kvantizacije kad simboli pripadaju ove sim-bol klase. Nalazimo najvei podprostor ultradistribucija za koje konvolucija sa gausovog jezgra postoji. To prua mogunost da pro-xirimo definiciju Anti-Wick kvantizacije za simbole koje nemoraju da su temperirane ultradistribucije.