Musielak-Orlicz Function Research Articles

Difference Sequence Spaces Defined by Musielak-Orlicz Function

Difference Sequence Spaces Defined by Musielak-Orlicz Function

Existence of renomalized solution for nonlinear elliptic boundary value problem without $$\Delta _{2}$$ -condition

In this paper we will prove in Musielak–Orlicz spaces, the existence of renomalized solution for nonlinear elliptic equations of Leray-Lions type, in the case where the Musielak–Orlicz function $$ \varphi $$ doesn’t satisfy the $$\Delta _{2}$$ condition while the right hand side f belongs to $$W^{-1} E_{\psi }(\Omega )$$ .

The Boundedness for Commutators of Anisotropic Calderón-Zygmund Operators

Let T be an anisotropic Calderon-Zygmund operator and φ : ℝn × [0, ∞) → [0, ∞) be an anisotropic Musielak-Orlicz function with φ(x, ·) being an Orlicz function and φ(·,t) being a Muckenhoupt A∞ (A) weight. In this paper, our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators. Precisely, when b ∈ BMOw(ℝn, A) (a proper subspace of anisotropic bounded mean oscillation space BMO(ℝn, A)), the commutator [b, T] is bounded from anisotropic weighted Hardy space Hl(ℝn, A) to weighted Lebesgue space Lw1 (ℝn) and when b ∈ BMO(ℝn) (bounded mean oscillation space), the commutator [b, T] is bounded on Musielak-Orlicz space Lφ(ℝn), which are extensions of the isotropic setting.

Some monotonicity properties in F-normed Musielak\u2013Orlicz spaces

Strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and their orthogonal counterparts are considered in the case of Musielak–Orlicz function spaces L^Phi (mu ) endowed with the Mazur–Orlicz F-norm as well as in the case of their subspaces E^Phi (mu ) with the F-norm induced from L^Phi (mu ). The presented results generalize some of the results from Cui et al. (Aequ Math 93:311–343, 2019) and Hudzik et al. (J Nonlinear Convex Anal 17(10):1985–2011, 2016), obtained only for Orlicz spaces as well as their subspaces of order continuous elements equipped with the Mazur–Orlicz F-norm.

Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Let (Ω,F,P) be a probability space, and let φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces WHφs(Ω), WHφM(Ω), WHφS(Ω), WPφ(Ω), and WQφ(Ω). We then use these atomic characterizations to obtain the boundedness of σ-sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of σ-sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.

On some spaces of Cesàro sequences of fuzzy numbers associated with λ-convergence and Orlicz function

Abstract In the present paper we shall introduce some generalized difference Cesàro sequence spaces of fuzzy real numbers defined by Musielak-Orlicz function and -convergence. We make an e ort to study some topological and algebraic properties of these sequence spaces. Furthermore, some inclusion relations between these sequence spaces are establish.

Multiplication between Hardy spaces and their dual spaces

For any p∈(0,1) and α=1/p−1, let Hp(Rn) and Cα(Rn) be the Hardy and the Campanato spaces on the n-dimensional Euclidean space Rn, respectively. In this article, the authors find suitable Musielak–Orlicz functions Φp, defined by setting, for any x∈Rn and t∈[0,∞),Φp(x,t):={t1+[t(1+|x|)n]1−pwhenn(1/p−1)∉N,t1+[t(1+|x|)n]1−p[log⁡(e+|x|)]pwhenn(1/p−1)∈N, and then establish a bilinear decomposition theorem for multiplications of functions in Hp(Rn) and its dual space Cα(Rn). To be precise, for any f∈Hp(Rn) and g∈Cα(Rn), the authors prove that the product of f and g, viewed as a distribution, can be decomposed into S(f,g)+T(f,g), where S is a bilinear operator bounded from Hp(Rn)×Cα(Rn) to L1(Rn) and T a bilinear operator bounded from Hp(Rn)×Cα(Rn) to the Musielak–Orlicz Hardy space HΦp(Rn) associated with the above Musielak–Orlicz function Φp. Such a bilinear decomposition is sharp when nα∉N, in the sense that any other vector space Y⊂HΦp(Rn) adapted to the above bilinear decomposition should satisfy (L1(Rn)+Y)⁎=(L1(Rn)+HΦp(Rn))⁎. To obtain the sharpness, the authors establish a characterization of the class of pointwise multipliers of Cα(Rn) by means of the dual space of HΦp(Rn), which is of independent interest. As an application, an estimate of the div-curl product involving the space HΦp(Rn) is discussed. This article naturally extends the known sharp bilinear decomposition of H1(Rn)×BMO(Rn).

Geometric difference of six-dimensional Riesz almost lacunary rough statistical convergence in probabilistic space of 𝜒𝑓 3

Abstract In this paper we study the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of {\chi^{3}} over probabilistic space P is defined by Musielak–Orlicz function. Since the study of convergence in Probabilistic space P is fundamental to probabilistic functional analysis, we feel that the concept of generalized geometric difference of six-dimensional rough almost lacunary statistical Cesáro of {\chi^{3}} over probabilistic space P defined by Musielak in a probabilistic space P would provide a more general framework for the subject.

New martingale inequalities and applications to Fourier analysis

Let (Ω,F,P) be a probability space and φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak–Orlicz space Lφ(Ω). Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on Lφ(Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for Lφ(Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces Hφs(Ω), Pφ(Ω), Qφ(Ω), HφS(Ω) and HφM(Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak–Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on HφS(Ω) and HφM(Ω), the authors obtain the Burkholder–Davis–Gundy inequality associated with Musielak–Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from Hφ[0,1) to Lφ[0,1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

On generalized geometric difference of six dimensional rough ideal convergent of triple sequence defined by Musielak-Orlicz function

On generalized geometric difference of six dimensional rough ideal convergent of triple sequence defined by Musielak-Orlicz function

Nonlinear elliptic equations with measure data in Musielak-Orlicz spaces

In this paper we will prove the existence of solutions of the unilateral problem
 Au - div Φ(x,u) + H(x, u, ∇ u) = μ
 in Musielak spaces, where A is a Leray-Lions operator defined on D(A) ⊂ W01 LM (Ω), μ ∈ L(Ω) + W-1 EM'(Ω), where M and M' are two complementary Musielak-Orlicz functions and both the first and the second lower terms Φ and H satisfies only the growth condition and u ≥ ζ where ζ is a measurable function.

Inclusion Theorem Between The Spaces Generated By Musielak-φ Function

Let X and G = (gi) respectively be a Banach space and a sequence of Musielak-φ function which is the generalization of Musielak-Orlicz function. Introducing the space of all X-valued sequences generated by Musielak-φ function, denoted by E∃(X,G), we investigate the sufficient and necessary condition for the inclusion relation of E∃(X,G) ⊂ F(X) and E(X) ⊂ E∃(X,G) where E ∈ {ℓ∞, ℓ1, c0} and F ∈ {ℓ∞, c0}.

Riesz triple probabilisitic of almost lacunary ces$\acute{A}$ro $C_{111}$ statistical convergence of $\chi^{3}$ defined by a Musielak Orlicz function

In this paper we study the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{3}$ over probabilistic $p-$ metric spaces defined by Musielak Orlicz function. Since the study of convergence in PP-spaces is fundamental to probabilistic functional analysis, we feel that the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{2}$ over probabilistic $p-$ metric spaces defined by Musielak in a PP-space would provide a more general framework for the subject.

Almost strongly Orlicz double sequence spaces of regular matrices and their applications to statistical convergence

In this paper, we introduce and study some strongly almost convergent double sequence spaces by Riesz mean associated with four-dimensional bounded regular matrix and a Musielak–Orlicz function over [Formula: see text]-normed spaces. We make an effort to study some topological and algebraic properties of these sequence spaces. We also study some inclusion relations between the spaces. Finally, we establish some relation between weighted lacunary statistical sequence spaces and Riesz lacunary almost statistical convergent sequence spaces over [Formula: see text]-normed spaces.

Musielak-Orlicz-Bochner function spaces which are uniformly noncreasy

In this paper, the criteria for uniform noncreasiness of Musielak–Orlicz–Bochner function spaces are given. Moreover authors also prove that the space LM(X) (resp LM0(X)) is uniformly noncreasy if and only if the space LM(X) (resp LM0(X)) is uniformly convex or uniformly smooth. As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given.

On Triple Sequence Spaces of Bernstein Operator of χ3 of Rough λ-Statistical Convergence in Probability of Random Variables Defined by Musielak-Orlicz Function

On Triple Sequence Spaces of Bernstein Operator of χ3 of Rough λ-Statistical Convergence in Probability of Random Variables Defined by Musielak-Orlicz Function

New real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz type

A real $n\times n$ matrix $A$ is called an expansive dilation if all of its eigenvalues $\lambda $ satisfy $|\lambda |\!\!>\!\!1$. Let $\varphi : \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. The aim of this article is to find an appropriate general space which includes the weak Hardy space of Fefferman and Soria, the weighted weak Hardy space of Quek and Yang}, the anisotropic weak Hardy space of Ding and Lan, the Musielak-Orlicz Hardy space of Ky and the anisotropic Hardy space of Musielak-Orlicz type of Li, Yang and Yuan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $m\in \mathbb {N}$ and obtain some new real-variable characterizations of $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ in terms of the radial, the non-tangential and the tangential maximal functions via a new monotone convergence theorem adapted to the weak anisotropic Musielak-Orlicz space $L^{\varphi , \infty }({\mathbb {R}}^n)$. These maximal function characterizations generalize the known results on the anisotropic weak Hardy space $H^{p, \infty }_A({\mathbb {R}}^n)$ with $p\in (0, 1]$ and are new even for their weighted variants or weak Orlicz-Hardy variants. As an application, the authors show the boundedness of a class of multilinear operators formed by the anisotropic Calderon-Zygmund operators from product weighted Lebesgue space to $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $\varphi (x,t):=t^p\omega (x)$ and $\omega \in \mathbb {A}_1(A)$, which is a weighted and non-isotropic extension of Grafakos.

Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals

AbstractIn the present paper we introduce generalized vector-valued Musielak-Orlicz sequence spacel(A,𝓜,u,p,Δr,∥·,… ,·∥)(X) and study some geometric properties like uniformly monotone, uniform Opial property for this space. Further, we discuss the operators ofs-type and operator ideals by using the sequence ofs-number (in the sense of Pietsch) under certain conditions on matrixA.

Riesz almost lacunary multiple triple sequence spaces of $\Gamma^{3}$ defined by a Musielak-Orlicz function

In this paper we introduce a new concept for Riesz almost lacunary $\Gamma^{3}$ sequence spaces strong $P-$ convergent to zero with respect to an Musielak-Orlicz function and examine some properties of the resulting sequence spaces. We also introduce and study statistical convergence of Riesz almost lacunary $\Gamma^{3}$ sequence spaces and also some inclusion theorems are discussed.

Property (k- $$\beta )$$ β ) of Musielak–Orlicz and Musielak–Orlicz–Cesàro spaces

In this paper, we investigate the geometric property (k- $$\beta )$$ for any fixed integer $$k\ge 1$$ of the space $$l_\varPhi ((E_{n}))$$ generated by a Musielak–Orlicz function $$\varPhi $$ and a sequence $$(E_n)$$ of finite dimensional spaces $$E_{n}$$ , $$n\in {\mathbb {N}}$$ , equipped with both the Luxemburg and the Amemiya norms. As a consequence, we obtain the property (k- $$\beta )$$ of Musielak–Orlicz–Cesaro spaces $$ces_\varPhi $$ using the approach recently considered by Saejung. Some applications to the Cesaro sequence spaces of order $$\alpha $$ and Cesaro difference sequence spaces of order m are also noted.