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.

Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball $T:=B_{\left|\right|·\left|\right|}(0,r)$, r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, $su{p}_{s,t\in T}|f\left(s\right)-f\left(t\right)|\le 6AB({\int }_0^r\psi (1/A{\epsilon }^{n-1}){\epsilon }^{n-1}d\epsilon +1/\left(n\right|B_{\left|\right|·\left|\right|}(0,1)\left|\right){\int }_T\phi (1/B||\nabla f\left(u\right)\left|\right|⁎\left)du\right)$, where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on...

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.

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 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...

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 ${\int }_0^{v₀}\varphi \left(u₀\right)·w<2$.

Let $\Omega \subset {ℝ}^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1{L}^n{log}^{\alpha }L\left(\Omega \right)$ into an Orlicz space corresponding to a Young function which behaves like $exp\left(t^{n/(n-1-\alpha )}\right)$ for large $t$. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle...

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.

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 $\varphi \in C⁰(ℝ,{ℝ}^N)$.

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces $L^{\Phi ,\kappa }\left(G\right)$ under conditions on $\Phi$ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals $\Phi (x,t)=t^{p\left(x\right)}+a\left(x\right)t^{q\left(x\right)}$, where $p(·)$ and $q(·)$ satisfy log-Hölder conditions and $a(·)$ is nonnegative, bounded and Hölder continuous.

We prove that the associate space of a generalized Orlicz space $L^{\phi (·)}$ is given by the conjugate modular ${\phi }^{*}$ even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling $\Phi$-function is equivalent to a doubling $\Phi$-function. As a consequence, we conclude that $L^{\phi (·)}$ is uniformly convex if $\phi$ and ${\phi }^{*}$ are weakly doubling.

Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator $T_f:u\mapsto u\circ f$ maps the Sobolev-Lorentz space $W{L}^{n,q}\left({\Omega }^{\text{'}}\right)$ to $W{L}^{n,q}\left(\Omega \right)$ for some q ≠ n then f must be a locally bilipschitz mapping.

We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which $\parallel {\sum }_{k=1}^n{\xi }_k{\parallel }_E\le C{n}^q$, where ${{\xi }_k}_{k\ge 1}\subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $exp\left(L_p\right)$, p ≥ 1. We further apply our results to the study of Banach-Saks...

For a function $f\in L_{loc}^p\left(ℝⁿ\right)$ the notion of p-mean variation of order 1, ${₁}^p(f,ℝⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1,p}\left(ℝⁿ\right)$ in terms of ${₁}^p(f,ℝⁿ)$ is directly related to the characterisation of $W^{1,p}\left(ℝⁿ\right)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces $H^{k,p}\left(G\right)$; $k\in {\mathbb{N}}^{*}$, $1\le p\le \infty$ and $G$ is the open unit disk or the annulus of the complex space $ℂ$.

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.