Real analytic parameter dependence of solutions of differential equations

Abstract

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation P (D)u = f , i.e., the question if for every family (f λ) ⊆ D (Ω) of distributions depending in a real analytic way on λ ∈ U , U a real analytic manifold, there is a family of solutions (u λ) ⊆ D (Ω) also depending analytically on λ such that P (D)u λ = f λ for every λ ∈ U,

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

“…The analysis of the Cousin problem requires the study of topological properties of the spaces of quasianalytic functions. The space of real analytic functions has been thoroughly investigated in the literature and its locally convex structure is by now well understood; see [3,12,13,23] for the scalar-valued case and [2,4,11,21] for the vectorvalued case. This is much less the case for other spaces of quasianalytic functions, although some work has been done [3,7,22,30].…”

“…This result enables us to give an explicit system of seminorms generating the topology on the quasianalytic function spaces of Roumieu type; such a projective topological description plays an important role in the analysis of the vector-valued case. We study vector-valued quasianalytic functions in Section 5, we closely follow there Komatsu's approach from [19]; in order to discuss their topological properties, we make use of the dual interpolation estimate for the space of real analytic functions [3] and a deep result of Domański on the ε-product of (P LS)-spaces [11]. The Cousin problem is solved in Section 6.…”

“…We are indebted to the authors of [3,11], as many of our proofs below rely on their results. In particular, Domański's work on the ε-product of (P LS)-spaces was very inspiring to us.…”

The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective

“…Examples for F are power series spaces Λ r (α); see [9,Section 29]. Moreover, in [3] surjectivity of the vector-valued operator P (D) : D (X, A (U )) → D (X, A (U )) is investigated, where A (U ) denotes the space of real analytic functions on a real analytic manifold U .…”

Weighted PLB-spaces of continuous functions arising as tensor products of a Fréchet and a DF-space

“…In particular there was an intense research on tensor products of PLS-spaces motivated by the problem of parameter dependence of solutions of PDE, see e.g. Bonet, Domański [10,12], Domański [15] and Piszczek [26]. We refer to the survey article [14] of Domański for applications, examples and further references.…”