Let G be a locally compact group with a fixed left Haar measure. Given Young functions φ and ψ, we consider the Orlicz spaces $L^{\phi }\left(G\right)$ and $L^{\psi }\left(G\right)$ on a non-unimodular group G, and, among other things, we prove that under mild conditions on φ and ψ, the set $(f,g)\in L^{\phi }\left(G\right)\times L^{\psi }\left(G\right):f*g$ is well defined on G is σ-c-lower porous in $L^{\phi }\left(G\right)\times L^{\psi }\left(G\right)$. This answers a question raised by Głąb and Strobin in 2010 in a more general setting of Orlicz spaces. We also prove a similar result for non-compact locally compact groups.

Criteria for compactly locally uniformly rotund points in Orlicz spaces are given.

A necessary and sufficient condition is given for a rearrangement invariant function space to contain a complemented isomorphic copy of l1(l2).

In this paper we give a sufficient condition on the pair of weights $(w,v)$ for the boundedness of the Weyl fractional integral $I_{\alpha }^{+}$ from $L^p\left(v\right)$ into $L^p\left(w\right)$. Under some restrictions on $w$ and $v$, this condition is also necessary. Besides, it allows us to show that for any $p:1\le p<\infty$ there exist non-trivial weights $w$ such that $I_{\alpha }^{+}$ is bounded from $L^p\left(w\right)$ into itself, even in the case $\alpha >1$.

Il lavoro presenta diverse caratterizzazioni degli spazi Lorentz-Zygmund generalizzati (GLZ) $L_{p,q;\alpha }\left(R\right)$, con $p,q\in \left(0,+\mathrm{\infty }\right]$, $m\in \mathbb{N}$, $\alpha \in {\mathbb{R}}^m$ e $\left(R,\mu \right)$ spazio misurato con misura $\mu \left(R\right)$ finita. Dato uno spazio misurato $\left(R,\mu \right)$ e $\alpha \in {\mathcal{R}}_{-}^m$ , otteniamo representazioni equivalenti per la (quasi-) norma dello spazio GLZ $L_{\mathrm{\infty },\mathrm{\infty };\alpha }\left(R\right)$. Inoltre, se $\left(R,\mu \right)$ è uno spazio misurato con misura finita e $\alpha \in {\mathcal{R}}_{+}^m$, viene presentata in termini di decomposizioni una norma equivalente per lo spazio $L_{1,1;\alpha }\left(R\right)$. Si dimostra che le norme equivalenti considerate per $L_{\mathrm{\infty },\mathrm{\infty };\alpha }\left(R\right)$, con $\left(R,\mu \right)$ uno spazio a misura finita, e la...

We give a description of the dual of a Calderón-Lozanovskiĭ intermediate space φ(X,Y) of a couple of Banach Köthe function spaces as an intermediate space ψ(X*,Y*) of the duals, associated with a "variable" function ψ.

Several conditions are given under which l1 embeds as a complemented subspace of a Banach space E if it embeds as a complemented subspace of an Orlicz space of E-valued functions. Previous results in Pisier (1978) and Bombal (1987) are extended in this way.

This paper is devoted to embedding theorems for classes of functions of several variables. One of our main objectives is to give an analysis of some basic embeddings as well as to study relations between them. We also discuss some methods in this theory that were developed in the last decades. These methods are based on non-increasing rearrangements of functions, iterated rearrangements, estimates of sections of functions, related mixed norms, and molecular decompositions.

We prove the equivalence of Nash type and super log Sobolev inequalities for Dirichlet forms. We also show that both inequalities are equivalent to Orlicz-Sobolev type inequalities. No ultracontractivity of the semigroup is assumed. It is known that there is no equivalence between super log Sobolev or Nash type inequalities and ultracontractivity. We discuss Davies-Simon's counterexample as the borderline case of this equivalence and related open problems.