Kernel theorems in spaces of generalized functions

Abstract : In analogy to the classical isomorphism between L(D(Rn);D0(Rm)) and D0(Rm+n) (resp. L(S(Rn); S0(Rm)) and S0(Rm+n)), we show that a large class of moderate linear mappings acting between the space GC(Rn) of compactly supported generalized functions and G(Rn) of generalized functions (resp. the space GS(Rn) of Colombeau rapidly decreasing generalized functions and the space G (Rn) of temperate ones) admits generalized integral representations, with kernels belonging to speci c regular subspaces of G(Rm+n) (resp. G (Rm+n)). The main novelty is to use accelerated -nets, which are unit elements for the convolution product in these regular subspaces, to construct the kernels. Finally, we establish a strong relationship between these results and the classical ones.
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Antoine Delcroix. Kernel theorems in spaces of generalized functions. Banach Center Publications, Institute of Mathematics Polish Academy of Sciences, 2010, 88, pp.77-89. ⟨10.4064/bc88-0-7⟩. ⟨hal-01529541⟩

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