• We study three problems:PROBLEM A. Let P(D) be a matrix of partial differential operators with constant coefficient and U an open, convex and bounded set in Rn. Consider the system P(D)u = f where f lies in a Sobolev space on Rn and satisfies the compatibility conditions in U. We give conditions for the existence of a solution u which is, roughly speaking, more regular than f by the degree of P(D) in the (global) Sobolev space meaning.PROBLEM B. Let P(D) and U be as above and let K be a compact convex subset of U. Let u be a function in a quasi-analytic class on U\ K such that P(D)u = 0 there. We give conditions on P(D) which imply that every such u may by continued to U as a solution of the considered homogeneous system and in the same quasi-analytic class. The conditions are also necessary.PROBLEM C. Let u be a (ultra-) distribution with compact support on Rn. The condition for u to act surjectively on a class of (ultra-distributions is usually expressed as a condition on the decrease of the Fourier transform of u. We localize this condition for the Gevrey distributions. We also prove that some measures of Cantor type, in particular the tenary measure, do not act surjectively on the usual distributions on R1.

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NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

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