Suppose E is fully symmetric Banach function space on (0,1) or (0,∞) or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on f ∈ E so that its orbit Ω(f) is the closed convex hull of its extreme points. We also give an application to symmetrically normed ideals of compact operators on a Hilbert space.

The space of all order continuous linear functionals on an Orlicz space $L^{\varphi }$ defined by an arbitrary (not necessarily convex) Orlicz function $\varphi$ is described.

Let L-phi be an Orlicz space defined by a Young function phi over a sigma-finite measure space, and let phi* denote the complementary function in the sense of Young. We give a characterization of the Mackey topology tau(L*,L-phi*) in terms of some family of norms defined by some regular Young functions. Next we describe order continuous (=absolutely continuous) Riesz seminorms on L-phi, and obtain a criterion for relative sigma(L-phi,L-phi*)-compactness in L-phi. As an application we get a representation...

We study order convexity and concavity of quasi-Banach Lorentz spaces ${\Lambda }_{p,w}$, where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that ${\Lambda }_{p,w}$ contains an order isomorphic copy of $l^p$. We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for ${\Lambda }_{p,w}$. We conclude with a characterization of the type and cotype of ${\Lambda }_{p,w}$ in the case when ${\Lambda }_{p,w}$ is a normable space.

Let ϕ and ψ be functions defined on [0,∞) taking the value zero at zero and with non-negative continuous derivative. Under very mild extra assumptions we find necessary and sufficient conditions for the fractional maximal operator $M_{\Omega }^{\alpha }$, associated to an open bounded set Ω, to be bounded from the Orlicz space $L^{\psi }\left(\Omega \right)$ into $L^{\varphi }\left(\Omega \right)$, 0 ≤ α < n. For functions ϕ of finite upper type these results can be extended to the Hilbert transform f̃ on the one-dimensional torus and to the fractional integral operator $I_{\Omega }^{\alpha }$, 0...

We study the sequence $u_n$, which is solution of $-div\left(a(x,𝔻u_n)\right)+{\Phi }^{\text{'}\text{'}}\left(\right|u_n\left|\right)u_n=f_n+g_n$ in $\Omega$ an open bounded set of ${𝐑}^N$ and $u_n=0$ on $\partial \Omega$, when $f_n$ tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the $N$-function $\Phi$, and prove a non-existence result.

We study the sequence un, which is solution of $-\mathrm{div}\left(a(x,\nabla u_n)\right)+{\Phi }^{\text{'}\text{'}}\left(\right|u_n\left|\right)u_n=f_n+g_n$ in Ω an open bounded set of RN and un= 0 on ∂Ω, when fn tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the N-function Φ, and prove a non-existence result.

Let $M$ be a von Neumann algebra, let $\varphi$ be a weight on $M$ and let $\Phi$ be $N$-function satisfying the $({\delta }_2,{\Delta }_2)$-condition. In this paper we study Orlicz spaces, associated with $M$, $\varphi$ and $\Phi$.

Let M be the Hardy-Littlewood maximal operator defined by:Mf(x) = supx ∈ Q 1/|Q| ∫Q |f| dx, (f ∈ Lloc(Rn)),where the supreme is taken over all cubes Q containing x and |Q| is the Lebesgue measure of Q. In this paper we characterize the Orlicz spaces Lφ*, associated to N-functions φ, such that M is bounded in Lφ*. We prove that this boundedness is equivalent to the complementary N-function ψ of φ satisfying the Δ2-condition in [0,∞), that is, sups&gt;0 ψ(2s) / ψ(s) &lt; ∞.

We prove basic properties of Orlicz-Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy-Littlewood maximal operator M from one Orlicz-Morrey space to another. For example, if f ∈ L(log L)(ℝⁿ), then Mf is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M, we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.