Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková
An extended version of the Maz'ya-Shaposhnikova theorem on the limit as $s\to 0^+$ of the Gagliardo-Slobodeckij fractional seminorm is established in the Orlicz space setting. Our result holds in fractional Orlicz-Sobolev spaces associated with Young functions satisfying the $Δ_2$-condition, and, as shown by counterexamples, it may fail if this condition is dropped.
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A new class of convex functions called functions, Young functions, strong Young functions and Orlicz functions are introduced by relaxing the definitions of functions, Young functions, strong Young functions and Orlicz functions. Then, new classes of Orlicz spaces and Orlicz Sobolev space will be defined.
We deal with Orlicz-Sobolev embeddings in open subsets of $\mathbb{R}^n$. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces with respect to a Frostman measure, and, in particular, for trace embeddings on the boundary.
Espaces d'Orlicz, Orlicz-Sobolev et application aux E-D-P
本文定义了Orlicz空间与Orlicz-Sobolev空间，研究其拓扑性质，并探讨其在偏微分方程中通过变分方法的应用。
In this paper we build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.
We prove capacitary strong type inequalities for functions belonging to Orlicz-Sobolev spaces. As an application we consider capacitary averages and their limits.
In this article we develop few of the analogous theoretical results of Clifford analysis over Orlicz-Sobolev spaces and study mapping properties of the Dirac operator and the Teodorescu transform over these function spaces. We also get analogous decomposition results of Clifford valued Orlicz spaces and the generalized Orlicz-Sobolev spaces.
We provide a necessary condition on the regularity of domains for the optimal embeddings of first order (and higher order) Orlicz-Sobolev spaces into Orlicz spaces in the sense of \cite{Cia96} (and \cite{Cia06}).
Composition operators in Orlicz-Sobolev spaces
研究 Orlicz-Sobolev 空间间 Nemytskii 复合算子的连续性，建立并改进经典 Sobolev 结果，且涵盖各向异性情形。
The aim in the present paper is to study removable sets for weighted Orlicz-Sobolev spaces. We generalize the definition of porous sets and show that the porous sets lying in a hyperplane are removable.
We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,φ}(\mathbb{R}^n)$. Under some natural assumptions on $φ$ we show that the maximal function is bounded and continuous in $W^{1,φ}(\mathbb{R}^n)$.
We study removable sets for the Orlicz-Sobolev space $W^{1,Ψ},$ for functions of the form $Ψ(t)=t^p\log^λ(e+t).$ We show that $(p,λ)$-porous sets lying in a hyperplane are removable and that this result is essentially sharp.
The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $\mathbb R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces, is also established. Both spaces of order $s\in (0,1)$, and higher-order spaces are considered. Related Hardy type inequalities are proposed as well. An extension theorem is proved, that enables us to derive embeddings for spaces defined in Lipschitz domains. Necessary and sufficient conditions for the compactness of fractional Orlicz-Sobolev…
In this note we study the limit as $s\downarrow 0$ of fractional Orlicz-Sobolev seminorms in Carnot groups. This closes the study started in [10]
In this paper we present a method for determining whether there exists a largest Orlicz space $L^A(Ω)$ satisfying the Sobolev embedding $W^mL^A(Ω) \to Y(Ω)$ where $Y(Ω)$ stands for an arbitrary so-called Marcinkiewicz endpoint space. The tool developed in this work enables us to investigate the optimality of Orlicz domain spaces in Sobolev embeddings and also in Sobolev trace embeddings on domains $Ω$ in $\mathbb{R}^n$ with various regularity.
In this short article we show a particular version of the Hedberg inequality which can be used to derive, in a very simple manner, functional inequalities involving Sobolev and Besov spaces in the general setting of Lebesgue spaces of variable exponents and in the framework of Orlicz spaces.
On the Continuous embeddings between the fractional Hajłasz-Orlicz-Sobolev spaces
本文在Orlicz函数满足特定条件时，证明了度量测度空间上分数阶Orlicz-Sobolev空间与分数阶Hajłasz-Orlicz-Sobolev空间之间的连续嵌入关系。
Under appropriate assumptions on the $N(Ω)$-fucntion, Morrey estimate is presented in the Musielak-Orlicz-Sobolev space.The assumptions include a new increasing condition on the $x$-derivative of the Young complementary function of the $N(Ω)$-fucntion. The conclusion applies to several important nonlinear examples frequently appeared in mathematical literature.
In this paper, we show that Orlicz--Sobolev spaces $W^{1,ϕ}(Ω)$ can be characterized with the ACL- and ACC-characterizations. ACL stands for absolutely continuous on lines and ACC for absolutely continuous on curves. Our results hold under the assumptions that $C^1(Ω)$ functions are dense in $W^{1,ϕ}(Ω)$, and $ϕ(x,β) \geq 1$ for some $β> 0$ and almost every $x \in Ω$. The results are new even in the special cases of Orlicz and double phase growth.
A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into $L^\infty(\mathbb R^n)$ fails. They also augment a classical embedding t…
We study the existence of an extension operator $Λ\colon W^{1,φ}(Ω)\to W^{1,ψ}(\mathbb{R}^n)$. We assume that $φ\in Φ_\mathrm{w}(Ω)$ has generalized Orlicz growth, $ψ\in Φ_\mathrm{w}(\mathbb{R}^n)$ is an extension of $φ$, and that $Ω\subset\mathbb{R}^n$ is an $(ε,δ)$-domain. Special cases include the classical constant exponent case, the Orlicz case, the variable exponent case, and the double phase case.
Fractional Orlicz-Sobolev embeddings into Campanato type spaces
该研究刻画了欧氏空间上分数阶Orlicz-Sobolev空间到各类Campanato型空间的最优嵌入，推导了BMO、VMO相关特例并指出其与经典分数阶Sobolev空间嵌入的差异。
On the compactness of embeddings for a class of weighted Orlicz-Sobolev sequence spaces
本文引入一类加权Orlicz-Sobolev序列空间，证明其连续性与紧性准则，连续性源于Orlicz函数逐点比较，紧性结合Schauder基与局部条件，并给出实例与开放问题。
The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to H.Lebesgue. It was introduced, in the setting of Sobolev spaces, by J.Manfredi, and thoroughly investigated in the more general framework of Orlicz-Sobolev spaces by diverse authors, including T.Iwaniec, J.Kauhanen, P.Koskela, J.Maly, J.Onninen, X.Zhong. The present paper complements and augments the available theory of pointwise regularity properties of weakly monotone functions in Orlicz-Sobolev spaces. In particular, a variant is proposed in a customary condition ensurin…
Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions
该研究针对形如$Φ(x,t)=t^{p(x)}(\log(e+t))^{q(x)}$的Musielak-Orlicz-Sobolev空间，探究其嵌入性质关于区域的充要条件。
On the modulus of continuity of fractional Orlicz-Sobolev functions
该研究给出分数阶Orlicz-Sobolev空间连续嵌入一致连续函数空间的充要条件与最优连续模，补充超临界Sobolev情形的嵌入结果，证明需采用新策略。
We prove the density of smooth functions in the modular topology in the Musielak-Orlicz-Sobolev spaces essentially extending the results of Gossez \cite{GJP2} obtained in the Orlicz-Sobolev setting. We impose new systematic regularity assumption on $M$ which allows to study the problem of density unifying and improving the known results in the Orlicz-Sobolev spaces, as well as the variable exponent Sobolev spaces. We confirm the precision of the method by showing the lack of the Lavrentiev phenomenon in the double-phase case. Indeed, we get the modular approximation of $W^{1,p}_0(Ω)$ functions…
Capacity and Hausdorff measure in Musielak-Orlicz-Sobolev spaces
该研究在 Musielak-Orlicz-Sobolev 空间中建立零容量集合与零广义 Hausdorff 测度之间的对应关系，分别针对 p(·)-容量及一类 Φ(·,·)-容量给出结论。
Strong and weak sharp bounds for Neural Network Operators in Sobolev-Orlicz spaces and their quantitative extensions to Orlicz spaces
本文研究了神经网络算子在Sobolev-Orlicz和Orlicz空间中的强弱sharp界，并引入了新的工具和方法，扩展到Orlicz空间的定量分析。
We compute the relaxation of the total energy related to a variational model for nematic elastomers, involving a nonlinear elastic mechanical energy depending on the orientation of the molecules of the nematic elastomer, and a nematic Oseen--Frank energy in the deformed configuration. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation belongs to an Orlicz-Sobolev space with an integrability just above the space dimension minus one, and does not present cavitation. We benefit from the fine properties of orientation-preserving map…
We extend the existence theorems in [Barchiesi, Henao \& Mora-Corral; ARMA 224], for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration, and a functional corresponding to the nematic energy (or the exchange and magnetostatic energies in magnetoelasticity) where the energy density is integrated over the deformed configuration. In order to obtain the desired compactness and lower semicontinu…
In this paper, based on concepts of convex sets and convex functions, we introduce new concepts of functions, Young functions, strong Young functions and Orlicz functions by relaxing definitions of functions, Young functions, strong Young functions, Orlicz functions respectively. We also give concepts of Orlicz spaces, weak Orlicz spaces, Orlicz Sobolev spaces, weak Orlicz Sobolev spaces, Orlicz Morrey spaces and weak Orlicz Morrey spaces, Orlicz Lorentz spaces and weak Orlicz Lorentz spaces. In addition, we consider the implicit properties based on the effect of an operator on the functions a…
The kernel of composition operator $C_T$ on Orlicz-Sobolev space is obtained. Using the kernel, a necessary and a sufficient condition for injectivity of composition operator $C_T$ has been established. Composition operators on Orlicz-Sobolev space with finite ascent as well as infinite ascent have been characterized.
In this paper, we investigate the boundedness of maximal operator and its commutators in generalized Orlicz-Morrey spaces on the spaces of homogeneous type. As an application of this boundedness, we give necessary and sufficient condition for the Adams type boundedness of fractional integral and its commutators in these spaces. We also discuss criteria for the boundedness of these operators in Orlicz spaces.
In the present paper, we will characterize the boundedness of the generalized fractional integral operators $I_ρ$ and the generalized fractional maximal operators $M_ρ$ on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces.
We show the density of smooth Sobolev functions $W^{k,\infty}(Ω)\cap C^\infty(Ω)$ in the Orlicz-Sobolev spaces $L^{k,Ψ}(Ω)$ for bounded simply connected planar domains $Ω$ and doubling Young functions $Ψ$.
By a stronger compact boundary embedding theorem in Musielak-Orlicz-Sobolev space developed in the paper, variational method is employed to deal with the nonlinear elliptic equation with the nonlinear Neumann boundary condition in the framework of Musielak-Orlicz-Sobolev space.
We investigate the commutators $[b,I_ρ]$ of generalized fractional integral operators $I_ρ$ with functions $b$ in generalized Campanato spaces and give a necessary and sufficient condition for the boundedness of the commutators on Orlicz spaces. To do this we define Orlicz spaces with generalized Young functions and prove the boundedness of generalized fractional maximal operators on the Orlicz spaces.
We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces. To do this we prove the weak-weak type modular inequality of the Hardy-Littlewood maximal operator with respect to the Young function. Orlicz-Morrey spaces contain $L^p$ spaces ($1\le p\le\infty$), Orlicz spaces and generalized Morrey spaces as special cases. Hence we get necessary and sufficient conditions on these function spaces as corollaries.
In this article we define a class of fractional Orlicz-Sobolev spaces on Carnot groups and, in the spirit of the celebrated results of Bourgain-Brezis-Mironescu and of Maz'ya-Shaposhnikova, we study the asymptotic behavior of the Orlicz functionals when the fractional parameter goes to $1$ and $0$.
In this paper, we solve the Dirichlet problem for Orlicz-Sobolev maps between singular metric spaces that extends the corresponding result of Guo et al. [arXiv 2021]. As an intermediate step, we develop a version of Rellich-Kondrachov compactness theorem for Orlicz-Sobolev mappings between metric spaces that extends a previous result of Guo and Wenger [Comm. Anal. Geom. 2020]. Another crucial ingredient is an Orlicz-Sobolev extension of the trace theory for metric valued Sobolev maps developed by Korevaar and Schoen [Comm. Anal. Geom. 1993].
In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation. To best of our knowledge, these results are also new in the special case of double phase spaces.
In the present paper, we shall give a characterization for weak/strong Adams-type boundedness of the fractional maximal operator on generalized Orlicz-Morrey spaces.
In this paper we define the fractional order Orlicz-Sobolev spaces, and prove its convergence to the classical Orlicz-Sobolev spaces when the fractional parameter $s\uparrow 1$ in the spirit of the celebrated result of Bourgain-Brezis-Mironescu. We then deduce some consequences such as $Γ-$convergence of the modulars and convergence of solutions for some fractional versions of the $Δ_g$ operator as the fractional parameter $s\uparrow 1$.
We study the fractional maximal operator acting between Orlicz spaces. We characterise whether the operator is bounded between two given Orlicz spaces. Also a necessary and sufficient conditions for the existence of an optimal target and domain spaces are established and the explicit formulas of corresponding optimal Young functions are presented.
In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator $M_α$ on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator $M_{b,α}$ and nonlinear commutator of fractional maximal operator $[b,M_α]$ on Orlicz spaces, when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.
Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces
本文研究张量场高阶链式法则，利用嵌套集合划分与广义贝尔多项式推广 Faà di Bruno 公式，并给出 Sobolev-Slobodeckij、Musielak-Orlicz 范数及全变差半范数估计。
Sobolev embeddings in Musielak-Orlicz space
本文提出了基于一般Musielak-Orlicz范数的Sobolev空间嵌入定理，通过引入最佳Sobolev共轭建立了Sobolev及Sobolev-Poincaré不等式，并利用Riesz位势的新型弱型不等式恢复并补充了多种标准空间在临界情况下的最佳嵌入。
The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz-Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.
We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions …
We investigate Sobolev spaces $W^{1,Φ}$ associated to Musielak-Orlicz spaces $L^Φ$. We first present conditions for the boundedness of the Voltera operator in $L^Φ$. Employing this, we provide necessary and sufficient conditions for $W^{1,Φ}$ to contain isomorphic subspaces to $\ell^\infty$ or $\ell^1$. Further we give necessary and sufficient conditions in terms of the function $Φ$ or its complementary function $Φ^*$ for reflexivity, uniform convexity, $B$-convexity and superreflexivity of $W^{1,Φ}$. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,φ}$ where …
Some notes on commutators of the fractional maximal and Riesz potential operators on Orlicz spaces
本文研究分数阶极大函数和Riesz位势在Orlicz空间上的交换子有界性，在更宽松条件下获得了新结果。
We study coercive inequalities in Orlicz spaces associated to the probability measures on finite and infinite dimensional spaces which tails decay slower than the Gaussian ones. We provide necessary and sufficient criteria for such inequalities to hold and discuss relations between various classes of inequalities.
We study a non-homogeneous boundary value problem in a smooth bounded domain in $\mathbb{R}^N$. We prove the existence of at least two nonnegative and non-trivial weak solutions. Our approach relies on Orlicz-Sobolev spaces theory combined with adequate variational methods and a variant of Mountain Pass Lemma.
We prove some general Sobolev-Orlicz, Nash and Faber-Krahn inequalities for positive operators of given ultracontractive norms of the spectral projectors on ]0, lambda]. For invariant operators on coverings of finite simplicial complexes this "ultracontractive spectral decay" is equivalent to von-Neumann's spectral density function. This allows in the polynomial decay case to relate the Novikov-Shubin numbers of such coverings to Sobolev inequalities on exact $\ell^2$-cochains, and to the vanishing of the torsion of the $\ell^{p,2}$-cohomology for some $p \geq 2$.
Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz-Morrey spaces
本文研究分数次积分算子交换子的端点估计，将Cruz-Uribe与Fiorenza关于Orlicz空间弱型有界性的结果推广至Orlicz-Morrey空间。
Generalized Sobolev-Orlicz spaces based on the Riesz fractional gradient as interpolation and potential spaces
本文证明基于Riesz分数阶梯度的Musielak-Orlicz函数分数阶Sobolev空间与广义Orlicz框架下的Bessel位势空间一致，并识别为复插值空间，进而获得空间的结构性质。
Inclusion relations among fractional Orlicz-Sobolev spaces and a Littlewood-Paley characterization
本文刻画了不同光滑性分数阶 Orlicz-Sobolev 空间的嵌入关系，并利用新的 Orlicz 卷积最优不等式建立了 Gagliardo-Slobodeckij 范数与 Littlewood-Paley 分解等范数的等价性，为嵌入证明提供关键工具。
Maz'ya-Shaposhnikova formula in Magnetic Fractional Orlicz-Sobolev spaces
本文证明 Maz'ya-Shaposhnikova 公式适用于磁性分数阶 Orlicz-Sobolev 空间，完善了已有关于 $s \uparrow 1$ 极限的研究。
Boundedness of commutator generated by fractional integral operator and Orlicz-BMO function
研究了分数积分算子与Orlicz-BMO函数生成的交换子从Orlicz-Hardy空间到Lebesgue空间的有界性，以及其在弱Lebesgue空间中的有界性。
In this article, the authors establish a new characterization of the Musielak--Orlicz--Sobolev space on $\mathbb{R}^n$, which includes the classical Orlicz--Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math. 19 (2017), 1650022, 13 pp] via weakening the assumption $f\in L^1(\mathbb R^n)$ into $f\in L_{\ma…
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