Soit $D$ un domaine borné strictement pseudoconvexe dans ${\mathbf{C}}^n$ à frontière régulière $\partial D$. On montre que tout compact d’une sous-variété $N$ de $\partial D$ dont l’espace tangent $T_p\left(N\right)$ en chaque point $p$ de $N$ est contenu dans le sous-espace complexe maximal de $T_p\left(\partial D\right)$ est un ensemble pic pour $A^{\infty }\left(D\right)$, la classe des fonctions analytiques dans $D$ dont toutes les dérivées sont continues dans $\bar{D}$.

Let X be an infinite-dimensional complex Banach space. Very recently, several results on the existence of entire functions on X bounded on a given ball B₁ ⊂ X and unbounded on another given ball B₂ ⊂ X have been obtained. In this paper we consider the problem of finding entire functions which are uniformly bounded on a collection of balls and unbounded on the balls of some other collection.

The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map $t{r}_{\Gamma }:B_{p₁,q}^s(ℝⁿ,w_{\varkappa }^{\Gamma })\to L_{p₂}\left(\Gamma \right)$, where Γ is a d-set with 0 < d < n and $w_{\varkappa }^{\Gamma }$ a weight of type $w_{\varkappa }^{\Gamma }\left(x\right)dist{(x,\Gamma )}^{\varkappa }$ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.

Let id be the natural embedding of the Sobolev space $W_p^l\left(\Omega \right)$ in the Zygmund space $L_q{\left(logL\right)}_a\left(\Omega \right)$, where $\Omega ={(0,1)}^n$, 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers $e_k\left(id\right)$ of this embedding and show that $e_k\left(id\right)\asymp k^{-\eta }$, where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.

We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

We discuss the existence of entropy solution for the strongly nonlinear unilateral parabolic inequalities associated to the nonlinear parabolic equations ∂u/∂t - div(a(x,t,u,∇u) + Φ(u)) + g(u)M(|∇u|) = μ in Q, in the framework of Orlicz-Sobolev spaces without any restriction on the N-function of the Orlicz spaces, where -div(a(x,t,u,∇u)) is a Leray-Lions operator and $\Phi \in C⁰(ℝ,{ℝ}^N)$. The function g(u)M(|∇u|) is a nonlinear lower order term with natural growth with respect to |∇u|, without satisfying the sign...

We prove an existence result of entropy solutions for a class of strongly nonlinear parabolic problems in Musielak-Sobolev spaces, without using the sign condition on the nonlinearities and with measure data.

We show that for "most" compact nonmetrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact nonmetrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.

One-sided versions of maximal functions for suitable defined distributions are considered. Weighted norm equivalences of these maximal functions for weights in the Sawyer's Aq+ classes are obtained.

We obtain the equivalence of the properties $\left(\beta \right)$ and (NUC) in Orlicz function spaces. This answers a question raised by Y. Cui, R. Pluciennik and T. Wang.