Soient $E$, $F$ des espaces de Banach $L_{\varphi }$, $L_{\psi }$ des espaces d’Orlicz, on définit les applications $\varphi -\psi$ sommantes de $E$ dans $F$. On montre que de telles applications sont $\varphi -\psi$ radonifiantes de $E$ dans $\sigma (F^{\text{'}\text{'}},F^{\text{'}})$.On donne une factorisation caractéristique des applications $\varphi -0$ sommantes.

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of $L^p$ and weak $L^p$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces $L^{\Phi }$ having the property $L^{\infty }\subset L^{\Phi }\subset L^p$, $1\le p<\infty$. The second contains spaces $L^{\Phi }$ that...

We discuss the convergence of approximate identities in Musielak-Orlicz spaces extending the results given by Cruz-Uribe and Fiorenza (2007) and the authors F.-Y. Maeda, Y. Mizuta and T. Ohno (2010). As in these papers, we treat the case where the approximate identity is of potential type and the case where the approximate identity is defined by a function of compact support. We also give a Young type inequality for convolution with respect to the norm in Musielak-Orlicz spaces.

Let G be a locally compact Hausdorff group with Haar measure, and let L⁰(G) be the space of extended real-valued measurable functions on G, finite a.e. Let ϱ and η be modulars on L⁰(G). The error of approximation ϱ(a(Tf - f)) of a function $f\in {\left(L⁰\left(G\right)\right)}_{\varrho +\eta }\cap DomT$ is estimated, where $\left(Tf\right)\left(s\right)={\int }_{G}K(t-s,f\left(t\right))dt$ and K satisfies a generalized Lipschitz condition with respect to the second variable.

We investigate the approximation properties of the partial sums of the Fourier series and prove some direct and inverse theorems for approximation by polynomials in weighted Orlicz spaces. In particular we obtain a constructive characterization of the generalized Lipschitz classes in these spaces.

Soient $W=L^{\text{'}}\left(\mu \right)$ et $V=C\left(X\right)$. Il existe une application (non linéaire) normiquement continue $T\mapsto P\left(T\right)$ de l’espace des opérateurs bornés de $W$ dans $V$ sur l’espace des opérateurs compacts (resp. faiblement compacts) de $W$ dans $V$ telle que $\parallel T-P\left(T\right)\parallel$ coïncide avec la distance de $T$ au sous-espace formé des opérateurs compacts (resp. faiblement compacts). Pour un opérateur donné $T$ de $W$ dans $V$ on étudie les propriétés de l’ensemble $\mathbf{K}\left(T\right)$ (resp. $\mathbf{F}\left(T\right)$) des opérateurs compacts (resp. faiblement compacts) tel que pour tout $\mathbf{R}$ de $\mathbf{K}\left(T\right)$ (resp. $\mathbf{K}\left(T\right)$) la quantité...

Let X denote the space of all real, bounded double sequences, and let Φ, φ, Γ be φ-functions. Moreover, let Ψ be an increasing, continuous function for u ≥ 0 such that Ψ(0) = 0.In this paper we consider some spaces of double sequences provided with two-modular structure given by generalized variations and the translation operator (...).

Let α be an isometric automorphism of the algebra ${}_p$ of bounded linear operators in ${}^p[0,1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].

We investigate a scale of $BM{O}_{\psi }$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BM{O}_{\psi }$-$L_{\infty }$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.

Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.