A nonlinear theory of infrahyperfunctions

Abstract

We develop a non-linear theory for infrahyperfunctions (also known as quasianalytic (ultra)distributions [25]). In the hyperfunction case our work can be summarized as follows: We construct a differential algebra that contains the space of hyperfunctions as a linear differential subspace and in which the multiplication of real analytic functions coincides with their ordinary product. Moreover, by proving an analogue of Schwartz's impossibility result for hyperfunctions, we show that this embedding is optimal. Our results fully solve a question raised by M. Oberguggenberger [34, p. 286, Prob. 27.2].

Rotation Invariant Ultradistributions

“…From now on we assume that M p satisfies (M.0), (M.1), (M.2) , and (QA). Our next considerations are in terms of sheaves of quasianalytic ultradistributions, we briefly discuss their properties following the approach from [5,10] (cf. [17] for hyperfunctions).…”

“…3.5], that is, for any * -quasianalytic functional there is a smallest * -carrier, say supp E * f , and one has supp A f = supp E * f . Hörmander only treats the Roumieu case in [10], but his proof can be modified to show the corresponding statement for the Beurling case [5,8].…”

“…Actually, in the Roumieu case the existence of B * can be established exactly as for hyperfunctions with the aid of Hörmander support theorem by using the Martineau-Schapira method [17]. Details for the Beurling case, which require a subtler treatment, will appear in the forthcoming paper [5]. Since it is important for us, we mention that on any bounded open set Ω the sections of B * are given by the quotient spaces The operation of taking spherical mean preserves the space E * [K] if K is a rotation invariant compact set.…”

Besov regularity in non-linear generalized functions

“…If one further restricts to nets of real valued constant functions, the resulting quotient algebra is R. The space of generalized functions G(Ω) then becomes a module over the ring C. We refer to [2,4,9] more details on Colombeau algebras. See also [5,6] for non-linear theories of ultradistributions and hyperfunctions.…”

Solution to the first Cousin problem for vector-valued quasianalytic functions

“…We shall also give sufficient conditions on a locally convex space F such that the Cousin problem is solvable in spaces of F -valued quasianalytic functions. We mention that in a forthcoming paper [9] the authors will apply the vectorvalued results from this article to construct sheaves of differential algebras in which the spaces of infrahyperfunctions of class {M p } [16] are embedded in such a way that the ordinary multiplication of ultradifferentiable functions of class {M p } is preserved.…”