Many of the known complemented subspaces of $L_p$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_p$. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions...

A super real closed ring is a commutative ring equipped with the operation of all continuous functions ℝⁿ → ℝ. Examples are rings of continuous functions and super real fields attached to z-prime ideals in the sense of Dales and Woodin. We prove that super real closed rings which are fields are an elementary class of real closed fields which carry all o-minimal expansions of the real field in a natural way. The main part of the paper develops the commutative algebra of super real closed rings, by...

In questo lavoro si studiano condizioni sufficienti sulla funzione peso $V$, espresse in termini di integrabilità, per la validità della disuguaglianza $${\left({\int }_B{u}^2\left(x\right)V\left(x\right)dx\right)}^{\frac{1}{2}}\le c{\left({\int }_B{\left(\nabla u\left(x\right)\right)}^{2}dx\right)}^{\frac{1}{2}},$$ dove $B$ denota una sfera in ${\mathbb{R}}^N$. Usando una tecnica di decomposizione di immersioni si dimostrano condizioni sufficienti in termini di appartenenza a spazi di Lebesgue, Lorentz-Orlicz e/o di tipo debole. Come applicazioni vengono fornite condizioni sufficienti per la proprietà forte di prolungamento unico per $\left|\mathrm{\Delta }u\right|\le V\left|u\right|$ nelle dimensioni 2 e 3.

We characterize the set of all functions f of R to itself such that the associated superposition operator Tf: g → f º g maps the class BVp1(R) into itself. Here BVp1(R), 1 ≤ p < ∞, denotes the set of primitives of functions of bounded p-variation, endowed with a suitable norm. It turns out that such an operator is always bounded and sublinear. Also, consequences for the boundedness of superposition operators defined on Besov spaces Bp,qs are discussed.

Soit ${\widetilde{𝒜}}_m$ l’algèbre des fonctions sur ${\mathbf{R}}^n$ engendrée par les fonctions polynomiales et les exponentielles de formes linéaires. La partie $S$ de ${\mathbf{R}}^n$ appartient à ${𝒫}_n$ si et seulement s’il existe $m$ et $F$ dans ${\widetilde{𝒜}}_{n+m}$ pour lesquels $S$ est l’image par la projection canonique de ${\mathbf{R}}^{n+m}$ sur ${\mathbf{R}}^n$, de l’ensemble des zéros de $F$. Soit ${\widetilde{𝒫}}_n$ le plus petit sous-ensemble de parties de ${\mathbf{R}}_n$ qui contient ${𝒫}_n$, l’adhérence de ses éléments et les images par la projection canonique de ${\mathbf{R}}^n$ qui contient ${𝒫}_n$, l’adhérence de ses éléments et les images par la...

Let D (resp. D*) be the subspace of C = C([0,1], R) consisting of differentiable functions (resp. of functions differentiable at the one point at least). We give topological characterizations of the pairs (C, D) and (C, D*) and use them to give some examples of spaces homeomorphic to CDor to CD*.

We generalize to some classes of ultradifferentiable jets or functions the classical Łojasiewicz Division Theorem and Glaeser Composition Theorem. The proof uses the desingularization results by Hironaka, Bierstone and Milman.