As is well known absolute convergence and unconditional convergence for series are equivalent only in finite dimensional Banach spaces. Replacing the classical notion of absolutely summing operators by the notion of 1 summing operators \[ \summ_k || Tx_k || \leq c || \summ_k e_k \otimes x_k ||_{\ell_1\otimes_{min}E}\] in the category of operator spaces, it turns out that there are quite different interesting examples of 1 summing operator spaces. Moreover, the eigenvalues of a composition $TS$ decreases of order $n^{\frac{1}{q}}$ for all operators $S$ factorizing completely through a commutative $C^*$-algebra if and only if the 1 summing norm of the operator $T$ restricted to a $n$-dimensional subspace is not larger than $c n^{1-\frac{1}{q}}$, provided $q>2$. This notion of 1 summing operators is closely connected to the notion of minimal and maximal operator spaces.
We study certain twisted sums of Orlicz spaces with non-trivial type which can be viewed as Fenchel-Orlicz spaces on ${\rm {\bf R}}^2$. We then show that a large class of Fenchel-Orlicz spaces on ${\rm {\bf R}}^n$ can be renormed to have property (M). In particular this gives a new construction of the twisted Hilbert space $Z_2$ and shows it has property (M), after an appropriate renorming.
We show that among all Musielak-Orlicz function spaces on a $σ$-finite non-atomic complete measure space equipped with either the Luxemburg norm or the Orlicz norm the only spaces with the Daugavet property are $L_1$, $L_{\infty}$, $L_1\oplus_1 L_{\infty}$ and $L_1\oplus_{\infty} L_{\infty}$. We obtain in particular complete characterizations of the Daugavet property in the weighted interpolation spaces, the variable exponent Lebesgue spaces (Nakano spaces) and the Orlicz spaces.
In this paper, we investige the concept of expansivity for composition operators on Orlicz-Lorentz spaces. We study necessary and sufficient conditions for expansivity, positive expansivity and uniformly expansivity for composition operators $C_τ$ on $\mathbb{L}^{φ,h}(μ)$. We extend the of results of [15] into Orlicz-Lorentz spaces
Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikodým property. Consequently it is shown that an Orlicz space $L_φ$ over a non-atomic $σ$-finite measure space $(Ω, Σ,μ)$, not necessarily separable, has the Radon-Nikodým property if and only if $φ$ is an $N$-function at infinity and satisfies the appropriate $Δ_2$ condition. For an Orlicz sequence space $\ell_φ$, it has the Radon-Nikodým property if and only if $φ$ satisfies condition $Δ_2^0$. In the second part the relationships between uniformly $\ell_1^2$ points of the unit sphere of a Bana…
Given an Orlicz $N$-function $φ$ and a positive decreasing weight $w$, we present criteria of the diameter two property and of the Radon-Nikodým property in Orlicz-Lorentz function and sequence spaces $Λ_{φ,w}$ and $λ_{φ,w}$. We show that in the spaces $Λ_{φ,w}$ or $λ_{φ,w}$ equipped with the Luxemburg norm, the diameter of any relatively weakly subset of the unit ball in these spaces is two if and only if $φ$ does not satisfy the appropriate $Δ_2$ condition, while they have the Radon-Nikodým property if and only if $φ$ satisfies the appropriate $Δ_2$ condition.
Let $1\le p\le\infty$. A Banach lattice $X$ is said to be $p$-disjointly homogeneous or $(p-DH)$ (resp. restricted $(p-DH)$) if every normalized disjoint sequence in $X$ (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in $X$ to the unit vector basis of $\ell_p$. We revisit $DH$-properties of Orlicz spaces and refine some previous results of this topic, showing that $(p-DH)$-property is not stable in the class of Orlicz spaces and the classes of restricted $(p-DH)$ and $(p-DH)$ Orlicz spaces are different. Moreover, we give a c…
In this paper we discuss the structure of Orlicz spaces and weak Orlicz spaces on $\mathbb{R}^n$. We obtain some necessary and sufficient conditions for the inclusion property of these spaces. One of the keys is to compute the norm of the characteristic functions of the balls in $\mathbb{R}^n$.
In this paper we extend the theory of Henstock-Orlicz spaces with respect to vector measure. We study the integral representation of operators. Lastly we study Uniformly convexity, reflexivity and the Radon-Nikodym property of the Henstock-Orlicz spaces
We improve our results on boundedness of the Riesz potential in the central Morrey-Orlicz spaces and the corresponding weak-type version. We also present two new properties of the central Morrey-Orlicz spaces: nontriviality and inclusion property.
In this paper, we study the Kadec-Klee property for convergence in measure of noncommutative Orlicz spaces $L_φ(\widetilde{\mathcal{M}},τ)$, where $\widetilde{\mathcal{M}}$ is a von Neumann algebra, and $φ$ is an Orlicz function. We show that if $φ\inΔ_{2}$, $L_φ(\widetilde{\mathcal{M}},τ)$ has the Kadec-Klee property in measure. As a corollary, the dual space and reflexivity of $L_φ(\widetilde{\mathcal{M}},τ)$ are given.
In this paper we show that Musielak--Orlicz spaces are UMD spaces under the so-called $Δ_2$ condition on the generalized Young function and its complemented function. We also prove that if the measure space is divisible, then a Musielak--Orlicz space has the UMD property if and only if it is reflexive. As a consequence we show that reflexive variable Lebesgue spaces $L^{p(\cdot)}$ are UMD spaces.
In this note we extend two characterizations of admissible operators with respect to $\mathrm{L}^p$ to more general Orlicz spaces. The equivalent conditions are given by the property that an associated operator generates a strongly continuous semigroup and in terms of a resolvent estimate.
We show that if $(\ell_{ϕ_0}, \ell_{ϕ_1})$ is a couple of suitable Orlicz sequence spaces then the corresponding Rochberg derived spaces of all orders associated to the complex interpolation method are Fenchel-Orlicz spaces. In particular, the induced twisted sums have the $(C[0, 1], \mathbb{C})$-extension property.
Let G be a locally compact group, let $Ω:G\times G\to \mathbb{C}^*$ be a 2-cocycle, and let $Φ$ be a Young function. In this paper, we consider the Orlicz space $L^Φ(G)$ and investigate its algebraic property under the twisted convolution $\circledast$ coming from $Ω$. We find sufficient conditions under which $(L^Φ(G),\circledast)$ becomes a Banach algebra or a Banach $*$-algebra; we call it a {\it twisted Orlicz algebra}. Furthermore, we study its harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, and having Wiener property, mostly for the case when…
Random variables equidistributed on convex bodies have received quite a lot of attention in the last few years. In this paper we prove the negative association property (which generalizes the subindependence of coordinate slabs) for generalized Orlicz balls. This allows us to give a strong concentration property, along with a few moment comparison inequalities. Also, the theory of negatively associated variables is being developed in its own right, which allows us to hope more results will be available. Moreover, a simpler proof of a more general result for $\ell_p^n$ balls is given.
Antilla, Ball and Perissinaki proved that the squares of coordinate functions in $\ell_p^n$ are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on R^n. From this, the concentration of the Euclidean norm and a form of the Central Limit Theorem for the generalized Orlicz balls is deduced. Also, a counterexample for the square negative correlation hypothesis for 1-symmetric bodies is given. Currently the CLT is known in full generality for convex bodies (see the paper "Power-law estimates for the central limit theorem for convex sets" by B. Klartag), w…
Necessary and sufficient conditions under which the Cesàro--Orlicz sequence space $\cfi$ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro--Orlicz spaces $\cfi$ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in $\cfi$ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces $\cfi$ are given.
Let $G$ be a locally compact group, $w$ be a weight on $G$ and $Φ$ be a Young function. We give some characterizations for translation operators to be topologically transitive and chaotic on the weighted Orlicz space $L_w^Φ(G)$. In particular, transitivity is equivalent to the blow-up/collapse property in our case. Moreover, the dense set of periodic elements implies transitivity automatically.
We study the boundedness of intrinsic square functions and their commutators on generalized Orlicz-Morrey spaces $M^{Φ,φ}(\mathbb{R}^n)$. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on weights $φ(x,r)$ without assuming any monotonicity property of $φ(x,r)$ on $r$.
We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces $M^{Φ,φ}_{w}({\mathbb R}^n)$. In all the cases, the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights $φ$ without assuming any monotonicity property of $φ(x,\cdot)$ with $x$ fixed.
In this paper we prove a plenty of new results concerning summabililty properties of multilinear mappings between Banach spaces, such as an extension of Littlewood's 4/3 Theorem. Among other features, it is shown that every continuous n-linear form on the disc algebra or the Hardy space is (1;2,...,2)-summing, the role of the Littlewood-Orlicz property in the theory is established and the interplay with almost summing multilinear mappings is explored.
Let $\boldΦ=(ϕ_n)$ be a Musielak-Orlicz function, $X$ be a real Banach space and $A$ be any infinite matrix. In this paper, a generalized vector-valued Musielak-Orlicz sequence space $l_{\bold Φ}^{A}(X)$ is introduced. It is shown that the space is complete normed linear space under certain conditions on the matrix $A$. It is also shown that $l_{\boldΦ}^{A}(X)$ is a $σ$- Dedikind complete whenever $X$ is so. We have discussed some geometric properties, namely, uniformly monotone, uniform Opial property for this space. Using the sequence of $s$-number (in the sense of Pietsch), the operators of…
For a measure space $Ω$ we extend the theory of Orlicz spaces generated by an even convex integrand $φ\colon Ω\times X \to \left[ 0, \infty \right]$ to the case when the range Banach space $X$ is arbitrary. Besides settling fundamental structural properties such as completeness, we characterize separability, reflexivity and represent the dual space. This representation includes the cases when $X'$ has no Radon-Nikodym property or $φ$ is unbounded. We apply our theory to represent convex conjugates and Fenchel-Moreau subdifferentials of integral functionals, leading to the first general such re…
We identify a large class of Orlicz spaces $X$ for which the topology $σ(X,X_n^\sim)$ fails the C-property introduced in [7]. We also establish a variant of the C-property and use it to prove a $w^*$-representation theorem for proper convex increasing functionals on dual Banach lattices that satisfy a suitable version of Delbaen's Fatou property. Our results apply, in particular, to risk measures on all Orlicz spaces over $[0,1]$ which is not $L_1[0,1]$.
Let G be a discrete group. Suppose that the reduced group C*-algebra of G is simple. We use results of Kalantar-Kennedy and Haagerup, and Banach space interpolation, to prove that, for p in (1,infinity), the reduced group L^p operator algebra F^p_r(G) and its *-analog B^{p,*}_r(G) are simple. If G is countable, we prove that the Banach algebras generated by the left regular representations on reflexive Orlicz sequence spaces and certain Lorentz sequence spaces are also simple. We prove analogous results with simplicity replaced by the unique trace property. For use in the Orlicz sequence space…
Perturbation Method in Musielak-Orlicz Sequence Spaces
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Faculty of Mathematics and Informatics, Sofia University St. Kliment Ohridski（索非亚圣克利门特奥赫里德斯基大学数学与信息学学院）；Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski（普罗夫迪夫帕伊西赫利安德斯基大学数学与信息学学院）
We investigate in a systematic way hypercontractivity property in Orlicz spaces for Markov semi-groups related to homogeneous and non homogeneous diffusions in $\mathbb{R}^{n}$. We provide an explicit construction of a family of Orlicz functions for which we prove that the associated hypercontractivity property is equivalent to a suitable functional inequality.
In this paper, we study the noncommutative Orlicz space $L_φ(\widetilde{\mathcal{M}},τ)$, which generalizes the concept of noncommutative $L^{p}$ space, where $\mathcal{M}$ is a von Neumann algebra, and $φ$ is an Orlicz function. As a modular space, the space $L_φ(\widetilde{\mathcal{M}},τ)$ possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace $E_φ(\widetilde{\mathcal{M}},τ)=\overline{\mathcal{M}\bigcap L_φ(\widetilde{\mathcal{M}},τ)}$ in $L_φ(\widetilde{\mathcal{M}},τ)$, which is closed under the norm topology and dense under the…
Let $Ω$ be a strongly Lipschitz domain of $\mathbb{R}^n$, whose complement in $\mathbb{R}^n$ is unbounded. Let $L$ be a second order divergence form elliptic operator on $L^2 (Ω)$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ have the Gaussian property $(G_{\mathrm{diam}(Ω)})$ with the regularity of their kernels measured by $μ\in(0,1]$, where $\mathrm{diam}(Ω)$ denotes the diameter of $Ω$. Let $Φ$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_Φ\in(n/(n+μ),1]$. In t…
The study of the Banach-Saks property in Banach spaces has a long and illustrious history. Of late, motivated by applications in financial mathematics, interest has arisen in the Banach-Saks type properties with respect to order convergence. This paper presents a study of order Banach-Saks properties in Banach function spaces, and in particular in rearrangement invariant spaces. Among the results obtained, we provide some sufficient conditions for the (weak) order Banach-Saks property. We also characterize the (weak) order Banach-Saks property in Orlicz spaces. It is also shown that the (weak)…
Let $Ω$ be either $\mathbb{R}^n$ or an unbounded strongly Lipschitz domain of $\mathbb{R}^n$, and $Φ$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_Φ\in(n/(n+1),1]$. Let $L$ be a divergence form elliptic operator on $L^2 (Ω)$ with the Neumann boundary condition and the heat semigroup generated by $L$ have the Gaussian property $(G_{\infty})$. In this paper, the authors introduce the Orlicz-Hardy space $H_{Φ,\,L}(Ω)$ via the nontangential maximal function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$…
For a singular measure $μ$, Ahlfors regular of order $α>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator $\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$ Here $P$ is the signed measure, $P=Vμ$ with density $V$ belonging to the Orlicz class $L^{Ψ,μ}$ with $Ψ(t)=(t+1)\log(t+1)-t.$ Using eigenvalue estimates for such operators, obtained in \texttt{arXiv:2011.14877}, we establish eigenvalue asymptotics of $\mathbf{T}(P,\mathbf{A})$ for a class of measures, including the ones suppor…
Let $L$ be the divergence form elliptic operator with complex bounded measurable coefficients, $ω$ the positive concave function on $(0,\infty)$ of strictly critical lower type $p_\oz\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors study the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$ and its dual space $\mathrm{BMO}_{ρ,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$. Several characterizations of $H_{ω,L}({\mathbb R}^n)$, including the molecular characterization, the Lusin-area function characte…
Let $Ω$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $ω\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (Ω)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $μ\in(0,1]$. Let $Φ$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_Φ^-\in(0,1]$. In this paper, the auth…
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex …
This work investigates spectrum and root functions (that is, eigen- and associated functions) of a Sturm-Liouville problem involving an abstract linear operator (nonselfadjoint in general) in the equation together with supplementary transmission conditions at the one interior singular point. So, the problem under consideration is not pure differential pro lem. At first we establish isomorphism and coerciveness with respect to the spectral parameter for the corresponding nonhomogeneous problem. Then by suggesting an own our method we prove that the spectrum of the considered problem is discrete…
In the paper we investigate automatic Fatou property of law-invariant risk measures on a rearrangement-invariant function space $\mathcal{X}$ other than $L^\infty$. The main result is the following characterization: Every real-valued, law-invariant, coherent risk measure on $\mathcal{X}$ has the Fatou property at every random variable $X\in \mathcal{X}$ whose negative tails have vanishing norm (i.e., $\lim_n\|X\mathbf{1}_{\{X\leq -n\}}\|=0$) if and only if $\mathcal{X}$ satisfies the Almost Order Continuous Equidistributional Average (AOCEA) property, namely, $\mathrm{d}(\mathcal{CL}(X),\mathc…
