The splitting problem is studied for short exact sequences consisting of countable projective limits of DFN-spaces (*) 0 → F → X → G → 0, where F or G are isomorphic to the space of distributions D'. It is proved that every sequence (*) splits for F ≃ D' iff G is a subspace of D' and that, for ultrabornological F, every sequence (*) splits for G ≃ D' iff F is a quotient of D'

We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near ℝ. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand-Shilov spaces $S{¹}_{\alpha }$ and $S{₁}^{\alpha }$ for α > 0 and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.

We prove the following common generalization of Maurey's extension theorem and Vogt's (DN)-(Omega) splitting theorem for Fréchet spaces: if T is an operator from a subspace E of a Fréchet space G of type 2 to a Fréchet space F of dual type 2, then T extends to a map from G into F'' whenever G/E satisfies (DN) and F satisfies (Omega).

We characterize all Fréchet quotients of the space (Ω) of (complex-valued) real-analytic functions on an arbitrary open set $\Omega \subseteq {ℝ}^d$. We also characterize those Fréchet spaces E such that every short exact sequence of the form 0 → E → X → (Ω) → 0 splits.

We study the structure of spaces of germs of holomorphic functions on compact sets in Fréchet spaces for (LB∞) as well as for (Ω,Ω).

Let s be the space of rapidly decreasing sequences. We give the spectral representation of normal elements in the Fréchet algebra L(s',s) of so-called smooth operators. We also characterize closed commutative *-subalgebras of L(s',s) and establish a Hölder continuous functional calculus in this algebra. The key tool is the property (DN) of s.

We prove precise estimates for the diametral dimension of certain weighted spaces of germs of holomorphic functions defined on strips near ℝ. This implies a full isomorphic classification for these spaces including the Gelfand-Shilov spaces $S{¹}_{\alpha }$ and $S{₁}^{\alpha }$ for α > 0. Moreover we show that the classical spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions are not isomorphic.

We characterize the partial differential operators P(D) admitting a continuous linear right inverse in the space of Fourier hyperfunctions by means of a dual (Ω̅)-type estimate valid for the bounded holomorphic functions on the characteristic variety $V_P$ near ${ℝ}^d$. The estimate can be transferred to plurisubharmonic functions and is equivalent to a uniform (local) Phragmén-Lindelöf-type condition.

The structure of the section space of a real analytic vector bundle on a real analytic manifold X is studied. This is used to improve a result of Grothendieck and Poly on the zero spaces of elliptic operators and to extend a result of Domański and the author on the non-existence of bases to the present case.

We give characterizations of certain properties of continuous linear maps between Fréchet spaces, as well as topological properties on Fréchet spaces, in terms of generalizations of Behrends and Kadets small ball property.

It is shown that complemented subspaces of s, that is, nuclear Fréchet spaces with properties (DN) and (Ω), which are 'almost normwise isomorphic' to a multiple direct sum of copies of themselves are isomorphic to s. This is applied, for instance, to spaces of Whitney jets on the Cantor set or the Sierpiński triangle and gives new results and also sheds new light on known results.

The paper gives a complete characterization of the subspaces, quotients and complemented subspaces of a stable power series space of infinite type without the assumption of nuclearity, so extending previous work of M. J. Wagner and the author to the nonnuclear case. Various sufficient conditions for the existence of bases in complemented subspaces of infinite type power series spaces are also extended to the nonnuclear case.

A Fréchet space with a sequence ${{\left|\right|·\left|\right|}_k}_{k=1}^{\infty }$ of generating seminorms is called tame if there exists an increasing function σ: ℕ → ℕ such that for every continuous linear operator T from into itself, there exist N₀ and C > 0 such that ${\left|\right|T\left(x\right)\left|\right|ₙ\le C\left|\right|x\left|\right|}_{\sigma \left(n\right)}$ ∀x ∈ , n ≥ N₀. This property does not depend upon the choice of the fundamental system of seminorms for and is a property of the Fréchet space . In this paper we investigate tameness in the Fréchet spaces (M) of analytic functions on Stein manifolds M equipped with the compact-open...

We discuss recent developments in the study of the homotopy classes for the Sobolev spaces $W^{1,p}\left(M;N\right)$. In particular, we report on the work of H. Brezis - Y. Li [5] and F.B. Hang - F.H. Lin [9].

The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $\left(L{B}^{\infty }\right)$ for the dual space of the space of germs of holomorphic functions on that compact set.

The linear isomorphism type of the tensor algebra T(E) of Fréchet spaces and, in particular, of power series spaces is studied. While for nuclear power series spaces of infinite type it is always s, the situation for finite type power series spaces is more complicated. The linear isomorphism T(s) ≅ s can be used to define a multiplication on s which makes it a Fréchet m-algebra $s_{•}$. This may be used to give an algebra analogue to the structure theory of s, that is, characterize Fréchet m-algebras...