Representation theorems for tempered ultradistributions

M. Budin\v cevi\'c, Z. Lozanov-Crvenkovi\'c and D. Peri\vsi\'c

Abstract: We consider classes of spaces of Beurling and Roumieu type tempered ultradistributions containing some spaces of quasianalytic tempered and all spaces of non-quasianalytic tempered ultradistributions. We prove that every ultradistribution $f$ in a space of the considered classes has the form $$ f = P(\Delta)u_{1} + u_{2}, $$ where $P$ is an ultradifferential operator, $u_{1} $ is a smooth function, $u_{2}$ is a real analytic function, and both of them satisfy some exponential growth conditions. Also, we give the boundary value representations for elements in the spaces of considered classes. Precisely, we prove that every solution of the heat equation, with appropriate exponential growth rate, defines an element in a space of the corresponding class, and conversely, that every element in a space of the considered classes is a boundary value of a solution of the heat equation with appropriate exponential growth rate.

Keywords: ultradistributions, tempered, boundary value, heat kernel

Classification (MSC2000): 46F12; 46F15

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