Considering jets, or functions, belonging to some strongly non-quasianalytic Carleman class on compact subsets of ${ℝ}^n$, we extend them to the whole space with a loss of Carleman regularity. This loss is related to geometric conditions refining Łojasiewicz’s “regular separation” or Whitney’s “property (P)”.

In this brief note, we see that if $A$ is a proper uniform algebra on a compact Hausdorff space $X$, then $A$ is flat.

For all convolution algebras L 1[0, 1); L loc1 and A(ω) = ∩n L 1(ωn), the derivations are of the form D μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D μ to a natural dual space is weak-star continuous.

An Orlicz-Pettis type property for vector measures and also the “Uniform Boundedness Principle” are shown to fail without local convexity assumption. The author asks under which generalized convexity hypotheses these properties remain true. This problem is expressed in terms of barrelledness type conditions.

Let n ≥ 2 and $H_k^{s,s^{\text{'}}}=u\in S^{\text{'}}\left({ℝ}^n\right):\parallel u{\parallel }_{s,s^{\text{'}}}<\infty$, where $\parallel u\parallel {²}_{s,s^{\text{'}}}={\left(2\pi \right)}^{-n}\int {(1+|\xi \left|²\right)}^s(1+|{\xi }^{\text{'}}{\left|²\right)}^{s^{\text{'}}}\left|Fu\left(\xi \right)\right|²d\xi$, $Fu\left(\xi \right)=\int e^{-ix\xi }u\left(x\right)dx$, ${\xi }^{\text{'}}\in {ℝ}^k$, k < n. We prove that for some s,s’ the space $H_k^{s,s^{\text{'}}}$ is a multiplicative algebra.

Nous montrons que $h^2(𝔻,L^1\left(𝕋\right))$ admet une norme équivalente $LUR,$ ce qui répond négativement à une question de Dowling, Hu et Smith. Puis nous obtenons une propriété de stabilité des opérateurs de Radon-Nikodym analytique. Motivés par l’identification entre $h^p(𝔻,X)$ et $V{B}^p(𝕋,X)$ où $X$ est un espace de Banach, pour un groupe abélien compact métrisable $G$, son dual $\Gamma$, et ${\Lambda }_2\subset {\Lambda }_1\subset \Gamma$, nous prouvons que, si l’espace $V{B}_{{\Lambda }_1}^p(G,X)/V{B}_{{\Lambda }_2}^p(G,X)$ a la propriété $Kadec-Klee-{\beta }^{\prime }-\omega$, alors il coincïde avec $L_{{\Lambda }_1}^p(G,X)/L_{{\Lambda }_2}^p(G,X),$$...$

This paper introduces the following definition: a closed subspace Z of a Banach space E is pseudocomplemented in E if for every linear continuous operator u from Z to Z there is a linear continuous extension ū of u from E to E. For instance, every subspace complemented in E is pseudocomplemented in E. First, the pseudocomplemented hilbertian subspaces of $L¹$ are characterized and, in $L^p$ with p in [1, + ∞[, classes of closed subspaces in which the notions of complementation and pseudocomplementation...

We introduce pseudodifferential operators (of infinite order) in the framework of non-quasianalytic classes of Beurling type. We prove that such an operator with (distributional) kernel in a given Beurling class ${}_{\left(\omega \right)}^{\text{'}}$ is pseudo-local and can be locally decomposed, modulo a smoothing operator, as the composition of a pseudodifferential operator of finite order and an ultradifferential operator with constant coefficients in the sense of Komatsu, both operators with kernel in the same class ${}_{\left(\omega \right)}^{\text{'}}$. We also...