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 $S_F^{\phi }$.

We use Simonenko quantitative indices of an $𝒩$-function $\Phi$ to estimate two parameters $q_{\Phi }$ and $Q_{\Phi }$ in Orlicz function spaces $L^{\Phi }[0,\infty )$ with Orlicz norm, and get the following inequality: $\frac{B_{\Phi }}{B_{\Phi }-1}\le q_{\Phi }\le Q_{\Phi }\le \frac{A_{\Phi }}{A_{\phi }-1}$, where $A_{\Phi }$ and $B_{\Phi }$ are Simonenko indices. A similar inequality is obtained in $L^{\Phi }[0,1]$ with Orlicz norm.

We prove the existence of solutions of the unilateral problem for equations of the type Au - divϕ(u) = μ in Orlicz spaces, where A is a Leray-Lions operator defined on $\left(A\right)\subset W₀¹L_M\left(\Omega \right)$, $\mu \in L¹\left(\Omega \right)+W^{-1}{E}_{M̅}\left(\Omega \right)$ and $\varphi \in C⁰(ℝ,{ℝ}^N)$.

Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that ${\Lambda }_{\varphi ,w}(0,\infty )$ (respectively, ${\Lambda }_{\varphi ,w}(0,1)$) is an order continuous Lorentz-Orlicz space. (1) ${\Lambda }_{\varphi ,w}$ has normal structure if and only if u₀ = 0 (respectively, ${\int }^{v₀}\varphi \left(u₀\right)·w<2andu₀<\infty ).$(2) ${\Lambda }_{\varphi ,w}$ has weakly normal structure if and only if ${\int }_0^{v₀}\varphi \left(u₀\right)·w<2$.

CONTENTSIntroduction............................................................................... 51. Definitions and preliminary results......................................... 72. Completeness of $L^{\Phi }(\mu ,\Re )$.............................. 93. Linear functionals on $L^{\Phi }(\mu ,\Re )$....................... 264. Geometry of Fenchel-Orlicz spaces........................................ 41References....................................................................................... 54

A new lower bound for the Jung constant $JC\left(l^{\left(\Phi \right)}\right)$ of the Orlicz sequence space $l^{\left(\Phi \right)}$ defined by an N-function Φ is found. It is proved that if $l^{\left(\Phi \right)}$ is reflexive and the function tΦ’(t)/Φ(t) is increasing on $(0,{\Phi }^{-1}\left(1\right)]$, then $JC\left(l^{\left(\Phi \right)}\right)\ge \left({\Phi }^{-1}(1/2)\right)/\left({\Phi }^{-1}\left(1\right)\right)$. Examples in Section 3 show that the above estimate is better than in Zhang’s paper (2003) in some cases and that the results given in Yan’s paper (2004) are not accurate.

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.

We will present relationships between the modular ρ* and the norm in the dual spaces $\left(L_{\Phi }\right)*$ in the case when a Musielak-Orlicz space $L_{\Phi }$ is equipped with the Orlicz norm. Moreover, criteria for extreme points of the unit sphere of the dual space $\left(L{⁰}_{\Phi }\right)*$ will be presented.

We study the canonical injection from the Hardy-Orlicz space $H^{\Psi }$ into the Bergman-Orlicz space ${}^{\Psi }$.

We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where $\Omega \subset {ℝ}^N$, N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $\rho \left(u\right)={\int }_{\Omega }\left(\Phi \right(\left|\nabla u\right|)+\Phi \left(\right|u\left|\right))dx$, M: [0,∞) → ℝ is a continuous function, $K\in L^{\infty }\left(\Omega \right)$, and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.

We study linear operators from a non-locally convex Orlicz space $L^{\Phi }$ to a Banach space $(X,|\left|·\right|{|}_X)$. Recall that a linear operator $T:L^{\Phi }\to X$ is said to be σ-smooth whenever $uₙ{⟶}^{\left(o\right)}0$ in $L^{\Phi }$ implies ${\left|\right|T\left(uₙ\right)\left|\right|}_X\to 0$. It is shown that every σ-smooth operator $T:L^{\Phi }\to X$ factors through the inclusion map $j:L^{\Phi }\to L^{\Phi ̅}$, where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators $T:L^{\Phi }\to X$. This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on...

We show that when the conjugate of an Orlicz function ϕ satisfies the growth condition Δ⁰, then the reflexive subspaces of $L^{\varphi }$ are closed in the L¹-norm. For that purpose, we use (and give a new proof of) a result of J. Alexopoulos saying that weakly compact subsets of such $L^{\varphi }$ have equi-absolutely continuous norm.

In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space $W^1{L}^{\Phi }\left([0,T]\right)$. We employ the direct method of calculus of variations and we consider  a potential  function $F$ satisfying the inequality $\left|\nabla F(t,x)\right|\le b_1\left(t\right){\Phi }_0^{\text{'}}\left(\right|x\left|\right)+b_2\left(t\right)$, with $b_1,b_2\in L^1$ and  certain $N$-functions ${\Phi }_0$.

We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with $L_{\infty }.$ We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair...

Let M₁ and M₂ be N-functions. We establish some combinatorial inequalities and show that the product spaces $\ell {ⁿ}_{M₁}\left(\ell {ⁿ}_{M₂}\right)$ are uniformly isomorphic to subspaces of L₁ if M₁ and M₂ are “separated” by a function $t^r$, 1 < r < 2.

Let G be a locally compact group, let (φ,ψ) be a complementary pair of Young functions, and let $L^{\phi }\left(G\right)$ and $L^{\psi }\left(G\right)$ be the corresponding Orlicz spaces. Under some conditions on φ, we will show that for a Banach $L^{\phi }\left(G\right)$-submodule X of $L^{\psi }\left(G\right)$, the multiplier space $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$ is a dual Banach space with predual $L^{\phi }\left(G\right)\bullet X:=\overline{span}ux:u\in L^{\phi }\left(G\right),x\in X$, where the closure is taken in the dual space of $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$. We also prove that if $\phi$ is a Δ₂-regular N-function, then $C{v}_{\phi }\left(G\right)$, the space of convolutors of $M^{\phi }\left(G\right)$, is identified with the dual of a Banach algebra of functions on G...

We obtain interpolation inequalities for derivatives: $\int M_{q,\alpha }\left(\right|\nabla f\left(x\right)\left|\right)dx\le C[\int M_{p,\beta }\left(\Phi ₁\right(x,\left|f\right|,|{\nabla }^{\left(2\right)}f\left|\right))dx+\int M_{r,\gamma }\left(\Phi ₂\right(x,\left|f\right|,|{\nabla }^{\left(2\right)}f\left|\right)\left)dx\right]$, and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ)$,$where ${\left|\right|·\left|\right|}_{(s,\kappa )}$ is the Orlicz norm relative to the function $M_{s,\kappa }\left(t\right)=t^s{\left(ln(2+t)\right)}^{\kappa }$. The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces...

We consider an embedding of the group of invertible transformations of [0,1] into the algebra of bounded linear operators on an Orlicz space. We show that if this embedding preserves the group action then the Orlicz space is an $L^p$-space for some 1 ≤ p < ∞.

We present several continuous embeddings of the critical Besov space $B_p^{n/p,\rho }\left(ℝⁿ\right)$. We first establish a Gagliardo-Nirenberg type estimate ${\left|\right|u\left|\right|}_{{Ḃ}_{q,w_r}^{0,\nu }}\le Cₙ{(1/(n-r))}^{1/q+1/\nu -1/\rho }{(q/r)}^{1/\nu -1/\rho }{\left|\right|u\left|\right|}_{{Ḃ}_p^{0,\rho }}^{(n-r)p/nq}{\left|\right|u\left|\right|}_{{Ḃ}_p^{n/p,\rho }}^{1-(n-r)p/nq}$, for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function $w_r\left(x\right)=1/{\left(\right|x|}^r)$ with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding $B_p^{n/p,\rho }\left(ℝⁿ\right)\hookrightarrow B_{\Phi ₀,w_r}^{0,\nu }\left(ℝⁿ\right)$, where the function Φ₀ of the weighted Besov-Orlicz space $B_{\Phi ₀,w_r}^{0,\nu }\left(ℝⁿ\right)$ is a Young function of the exponential type. Another point of interest is to embed $B_p^{n/p,\rho }\left(ℝⁿ\right)$ into the weighted Besov...

A real-valued Hardy space $H{¹}_{\surd }\left(\right)\subseteq L¹\left(\right)$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H{¹}_{\surd }\left(\right)$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H{¹}_{\surd }\left(\right)$, and no Orlicz...

Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E{*}_{\omega *}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G\left(x\right):={\int }_{\Omega }g(s,x\left(s\right))d\mu \left(s\right)$. Consider the integral functional G defined on some non-$L^p$-type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...

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...

We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from $d{*}_{(}w,1)$ into d(w,1), where $d{*}_{(}w,1)$ is a predual of a complex Lorentz sequence space d(w,1), if and only if wi ∈ c₀∖ℓ₂.

We characterize compact composition operators acting on weighted Bergman-Orlicz spaces ${}_{\alpha }^{\psi }=f\in H\left(\right):{\int }_{}\psi \left(\right|f\left(z\right)\left|\right)d{A}_{\alpha }\left(z\right)<\infty$, where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition $li{m}_{t\to \infty }\psi \left(t\right)/t=\infty$ and the Δ₂-condition. In fact, we prove that $C_{\phi }$ is compact on ${}_{\alpha }^{\psi }$ if and only if it is compact on the weighted Bergman space ${²}_{\alpha }$.

We study imbeddings of the Sobolev space $W^{m,\varrho }\left(\Omega \right)$: = u: Ω → ℝ with $\varrho ({\partial }^{\alpha }u/\partial x^{\alpha })$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, ${\tau }_{\varrho }$ and ${\sigma }_{\varrho }$, that are optimal with respect to the inclusions $W^{m,\varrho }\left(\Omega \right)\subset W^{m,{\tau }_{\varrho }}\left(\Omega \right)\subset L_{{\sigma }_{\varrho }}\left(\Omega \right)$. General formulas for ${\tau }_{\varrho }$ and ${\sigma }_{\varrho }$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

CONTENTSPreface..............................................................................................................................4Introduction........................................................................................................................51. Orlicz spaces..................................................................................................................6 1.1. Orlicz functions...........................................................................................................6 1.2....

Let $L:=-\Delta +V$ be a Schrödinger operator on ${ℝ}^n$ with $n\ge 3$ and $V\ge 0$ satisfying ${\Delta }^{-1}V\in L^{\infty }\left({ℝ}^n\right)$. Assume that $\varphi :{ℝ}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,·)$ is an Orlicz function, $\varphi (·,t)\in {𝔸}_{\infty }\left({ℝ}^n\right)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on ${ℝ}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}\left({ℝ}^n\right)\ni f\mapsto wf\in H_{\varphi }\left({ℝ}^n\right)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}\left({ℝ}^n\right)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }\left({ℝ}^n\right)$ under some assumptions on $\varphi$. As applications, the author further...