Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...

The aim of this paper is to establish the theorem of atomic decomposition of weighted Bergman spaces Ap(Ω), where Ω is a domain of finite type in C2. We construct a kernel function H(z,w) which is a reproducing kernel for Ap(Ω) and we prove that the associated integral operator H is bounded in Lp(Ω).

Nous dirons qu’un faisceau de groupes abéliens $ℱ$ sur un espace topologique $X$ est souple si, $\Omega$ étant un ouvert de $X$, $F_1$ et $F_2$ des fermés de $\Omega$, toute section de $ℱ$ sur $\Omega$ à support dans $F_1\cup F_2$ est somme de sections à support dans $F_1$ et $F_2$. Soit $M$ une variété analytique réelle, $S^{*}M$ son fibré cotangent en sphères, $C^f$ le faisceau sur $S^{*}M$ des microfonctions qui proviennent localement sur $S^{*}M$, de distributions. Nous montrons que le faisceau $C^f$ est souple. En particulier le faisceau ${𝒟}^{\text{'}}/𝒜$ sur $M$, quotient des distributions par...

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.The main results of the paper are: Theorem 1. Let a Banach space E be decomposed into a direct sum of separable and reflexive subspaces. Then for every Hausdorff locally convex topological vector space Z and for every linear continuous bijective operator T : E → Z, the inverse T^(−1) is a Borel map. Theorem 2. Let us assume the continuum hypothesis. If a Banach space E cannot...

Let $\Theta :={\theta }_I^e:e\in E,I\in D$ be a decomposition system for $L₂\left({ℝ}^d\right)$ indexed over D, the set of dyadic cubes in ${ℝ}^d$, and a finite set E, and let $\Theta ̃:=\Theta {̃}_I^e:e\in E,I\in D$ be the corresponding dual functionals. That is, for every $f\in L₂\left({ℝ}^d\right)$, $f={\sum }_{e\in E}{\sum }_{I\in D}⟨f,\Theta {̃}_I^e⟩{\theta }_I^e$. We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨f,\Theta {̃}_I^e⟩$, e ∈ E, I ∈ D. Typical examples of such decomposition systems...

Dans ce travail, nous étudions le problème de décomposicion suivant: Étant donnés deux ouverts bornés de Cp, Ω1 et Ω2 (vérifiant certaines conditions) et étant donnée une matrice A(z), carrée d'ordre n, dont les coefficients sont des fonctions holomorphes dans Ω1 ∩ Ω2, ayant une prolongement C∞ à l'adhérence (Ω1 ∩ Ω2), peut-on trouver deux matrices A1(z), A2(z) holomorphes dans Ω1 et Ω2 respectivement et se prolongeant de manière C∞ à (Ω1) et (Ω2) telles que sur Ω1 ∩ Ω2 on aitA = A1A2.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different subsets,...

We define tensor product decompositions of E₀-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent.

There have been recent attempts to develop the theory of Sobolev spaces $W^{1,p}$ on metric spaces that do not admit any differentiable structure. We prove that certain definitions are equivalent. We also define the spaces in the limiting case $p=1$.