Ideal Statistically Pre-Cauchy Triple Sequences of Fuzzy Number and Orlicz Functions

In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. It is shown that x = {xijk} is ideal statistically pre-Cauchy if and only if D (xijk, xpqr) ≥ ε, i ≤ m, j ≤ n, t ≤ k}| ≥ δ} ∈I. At the same time, we have proved x = {xijk} is ideal statistically convergent to x0 if and only if . Also, some properties of these new sequence spaces are investigated.

On some difference sequence spaces defined by a sequence of Orlicz functions

The idea of difference sequence spaces was introduced in （Klzmaz, 1981） and this concept was generalized in （Et and Colak, 1995）. In this paper we define some difference sequence spaces by a sequence of Orlicz functions and establish some inclusion relations.

A SEQUENCE SPACE DEFINED BY ORLICZ FUNCTIONS

In this paper we define a sequence space using Orlicz functions. We give certain properties and inclusion relations between known sequence spaces and new sequence space.

SOME DOUBLE SEQUENCE SPACES OF FUZZY NUMBERS DEFINED BY ORLICZ FUNCTIONS

In this article, we introduce some double sequence spaces of fuzzy real numbers defined by Orlicz function, study some of their properties like solidness, symmetricity, completeness etc, and prove some inclusion results.

Some I-Convergent Sequence Spaces Defined by Orlicz Functions

In this article we introduce the sequence spaces cI（M）, c0I（M）, mI（M） and m0I（M） using the Orlicz function M. We study some of the properties like solid, symmetric, sequence algebra, etc and prove some inclusion relations.

On a Class of Generalized Lacunary Difference Sequence Spaces Defined by Orlicz Functions

In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞ using m^(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞ and the Cesàro summable and strongly Cesàro summable sequences.

PADE APPROXIMANTS AS LIMITS OF RATIONAL FUNCTIONS OF BEST APPROXIMATION IN ORLICZ SPACE

In this paper, we prove that the best rational approximation of a given analytic function in Orlicz space L~*(G), where G = {|z|≤∈}, converges to the Pade approximants of the function as the measure of G approaches zero.

Direct and Reverse Carleson Conditions on Generalized Weighted Bergman-Orlicz Spaces

Let D be the open unit disk in the complex plane C. For a〉 -1, let dAa（z）=（1 ＋a） （1 -｜z}^2） ^a da（z）be the weighted Lebesgue measure on ]D. For a positive function ω ∈ L^1（D,dAa）, the generalized weighted Bergman-Orlicz spaceA^ψω（D,dAa）is｜｜f｜｜ω^ψ=∫Dψ｜F（z）｜ω（z）dA^（z） 〈 ∞,where q; is a strictly convex Orlicz function that satisfies other technical hypotheses. Let G be a measurable subset of D, we say G satisfies the reverse Carleson condition for A^ψω （D, dAa） if there exists a positive constant C such that ∫Gψ（f（z））ω（z）dAa（z）≥C∫Dψ（｜f（z）dAa（z）.for all f ∈ .A^ψω （D,dAa）. Let μ be a positive Borel measure, we say μ satisfies the direct Carleson condition if there exists a positive constant M such that for all f∈Aψ^ω （D,dAa）,∫Dψ（｜f（z）｜）dμ（z）≤M∫Dψ（｜f（z）｜）ω（z）dAa（a）.In this paper, we study the direct and reverse Carleson condition on the generalized weighted Bergman-Orlicz space Aω^ψ（D,dAa）.We present conditions on the set G such that＇the reverse Carleson condition＇holds. ＂Moreover, we give a sufficient condition for the finite positive Borel measure μ to satisfy the direct carleson condition on the generalized weighted Bergman-Orlicz spaces.

Noncommutative martingale Hardy-Orlicz spaces:Dualities and inequalities

We investigate dualities and inequalities related to noncommutative martingale Hardy-Orlicz spaces.More precisely,for a concave Orlicz functionΦ,we characterize the dual spaces of noncommutative martingale Hardy-Orlicz spaces HΦc(R)and hΦc(M),where R denotes a hyperfinite finite von Neumann algebra and M is a finite von Neumann algebra.The first duality is new even for classical martingales,which partially answers the problem raised by Conde-Alonso and Parcet(2016).We establish as well asymmetric martingale inequalities associated with Orlicz functions that are p-convex and q-concave for 0<p≤q<2.

Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates

Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a positive, continuous and strictly increasing function on （0, ∞）, which is of strictly critical lower type pФ （n/（n ＋ θ（L））, 1]. Denote by HФ, L（R^n） the Orlicz-Hardy space introduced in Jiang, Yang and Zhou＇s paper in 2009. If Ф is additionally of upper type 1 and subadditive, the authors then show that the Littlewood-Paley g-function gL maps HФ, L（R^n） continuously into LФ（R^n） and, moreover, the authors characterize HФ, L（R^n） in terms of the Littlewood-Paley gλ^＊-function with λ ∈ （n（2/pФ ＋ 1）, ∞）. If Ф is further slightly strengthened to be concave, the authors show that a generalized Riesz transform associated with L is bounded from HФ, L（R^n） to the Orlicz space L^Ф（R^n） or the Orlicz-Hardy space HФ （R^n）; moreover, the authors establish a new subtle molecular characterization of HФ, L （R^n） associated with L and, as applications, the authors then show that the corresponding fractional integral L^-γ for certain γ∈ E （0,∞） maps HФ, L（R^n） continuously into HФ, L（R^n）, where Ф satisfies the same properties as Ф and is determined by Ф and λ and also that L has a bounded holomorphic functional calculus in HФ, L（R^n）. All these results are new even when Ф（t） = t^p for all t ∈ （0, ∞） and p ∈ （n/（n ＋ θ（L））, 1].

Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L 2 (? n ) and generate an analytic semigroup {e ?tL }t?0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0,∞) be of upper type 1 and of critical lower type p o (ω) ? (n/(n+θ(L)),1] and ρ(t) = t t1/ω ?1(t ?1) for t ∈ (0,∞). We introduce the Orlicz-Hardy space H ω, L (? n ) and the BMO-type space BMO ρ, L (? n ) and establish the John-Nirenberg inequality for BMO ρ, L (? n ) functions and the duality relation between H ω, L ((? n ) and BMO ρ, L* (? n ), where L* denotes the adjoint operator of L in L 2 (? n ). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMO ρ, L* (? n ) and the molecular characterization of H ω, L (? n ); the latter is used to establish the boundedness of the generalized fractional operator L ρ ?γ from H ω, L (? n ) to H L 1 (? n ) or L q (? n ) with certain q > 1, where H L (? n ) is the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = t p for t ∈ (0,∞) and p ∈ (n/(n + θ(L)), 1].

Molecular Characterization of Anisotropic Musielak–Orlicz Hardy Spaces and Their Applications

Let A be an expansive dilation on Rn and φ ： Hn×[0, ∞）→[0, ∞） an anisotropic Musielak-Orlicz function. Let HAφ（R^n） be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of HAφ（R^n）. The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case （namely, A ：= 2In×n） coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space HP（R^n） with p∈[（0, 1] （in this case, A ：= 2In×n,φ（x, t） ：= t^p for all x ∈ R^n and t∈[0,∞））, this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Caldeon-Zygmund operators from HA^φ（Hn） to L^φ（R^n） or from HA^φ（Hn） to itself.

Composition of Fractional Orlicz Maximal Operators and A1-weights on Spaces of Homogeneous Type

For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type （X, d, μ） by Mα,θf（x） = supx∈（B）α ｜｜f｜｜θ,B, where ｜｜f｜｜θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈（0,1） then （Mα,θf）δ is an A1-weight.

Upper(Lower) Monotone Coefficient of a Point in Orlicz Function Spaces

The calculation expressions of upper （lower） monotone coefficient of a point in Orlicz function spaces are given.