We introduce the Musielak-Orlicz space of multifunctions $X_{m,\phi }$ and the set $S_F^{\phi }$ of φ-integrable selections of F. We show that some decomposable sets in Musielak-Orlicz space belong to $X_{m,\phi }$. We generalize Theorem 3.1 from [6]. Also, we get some theorems on the space $X_{m,\phi }$ and the set $S_F^{\phi }$.

Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...

Norm-to-weak* continuity of excess demand as a function of prices is proved by using our two-topology variant of Berge's Maximum Theorem. This improves significantly upon an earlier result that, with the extremely strong finite topology on the price space, is of limited interest, except as a vehicle for proving equilibrium existence. With the norm topology on the price space, our demand continuity result becomes useful in applications of equilibrium theory, especially to problems with continuous...

In this text, we present two recent results on the characterization of the lack of compactness of some critical Sobolev embedding. The first one derived in [5] deals with an abstract framework including Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. The second one established in [3] concerns the lack of compactness of $H^1\left({ℝ}^2\right)$ into the Orlicz space. Although the two results are expressed in the same manner (by means of defect measures) and rely on the defect of compactness due to concentration...

Assume that L p,q, $L^{p_1,q_1},...,L^{p_n,q_n}$ are Lorentz spaces. This article studies the question: what is the size of the set $E=\{(f_1,...,f_n)\in L^{p_{1,}{q}_1}\times \cdots \times L^{p_n,q_n}:f_1\cdots f_n\in L^{p,q}\}$. We prove the following dichotomy: either $E=L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$ or E is σ-porous in $L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E=L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$ or E is meager. This is a generalization of the results for classical L p spaces.

We survey recent dimension-invariant imbedding theorems for Sobolev spaces.

It is studied when inclusions between rearrangement invariant function spaces on the interval [0,∞) are disjointly strictly singular operators. In particular suitable criteria, in terms of the fundamental function, for the inclusions $L¹\cap L^{\infty }\hookrightarrow E$ and $E\hookrightarrow L¹+L^{\infty }$ to be disjointly strictly singular are shown. Applications to the classes of Lorentz and Marcinkiewicz spaces are given.