We prove that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to $\ell^1\cap\ell^Φ_{ϕ,w}$ (resp. to $\ell^1\cap\ell^Ψ_{ψ,\varrho}$), where $\ell^Φ_{ϕ,w}$ and $\ell^Ψ_{ψ,\varrho}$ are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences \[ ϕ=\{ϕ_n\},\quad ψ=\{ψ_n\},\quad w=\{w_n\},\quad \varrho=\{\varrho_n\} \] increase and satisfy the $Δ_2$-condition.
In this article we investigate the Fourier series and transforms for the functions defined on the [-pi, pi]^ d or on the R^d and belonging to the (Bilateral) Grand Lebesgue Spaces. As a particular case we obtain some results about Fourier's transform in the so-called exponential Orlicz spaces. We construct also several examples to show the exactness of offered estimations.
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
Using recent characterizations of the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces on the ball, we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H^{\infty}. Then, we prove that, for each Koranyi region Γ, there exists a map ϕtaking the ball into Γsuch that, C_ϕ is not compact on H^ψ\left(\mathbb{B}_{N}\right), when ψgrows fast. Finally, we give another characterization of the compactness of composition operator on weighted Bergman-Orlicz spaces in terms of an Orlicz "Angular derivat…
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.
Let $(Ω,Σ,μ)$ be a $σ$-finite complete measure space, $τ:Ω\rightarrowΩ$ be a measurable transformation and $ϕ$ be an Orlicz function. In this article, first a necessary and sufficient condition for the bounded multiplication operator $M_u$ on Orlicz space $L^ϕ(Ω)$ induced by measurable function $u$ to be completely continuous has been established. Next by using Radon-Nikodym derivative $ω= \frac{d μ\circ τ^{-1}}{d μ}$, the multiplication operator $M_u$ on weighted Orlicz space $L^ϕ_ω(Ω)$ have been characterized.
In this paper we discuss the structure of weighted weak Lebesgue spaces and weighted weak Orlicz spaces on $\mathbb{R}^n$. First, we present sufficient and necessary conditions for inclusion relation between weighted weak Lebesgue spaces. Next, we also obtain similar results on weighted weak Orlicz spaces. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in $\mathbb{R}^n$.
Let G be a locally compact abelian group, $ω:G\to (0,\infty)$ be a weight, and ($Φ$,$Ψ$) be a complementary pair of strictly increasing continuous Young functions. We show that for the weighted Orlicz algebra $L^Φ_ω(G)$, the weak amenability is obtained under conditions similar to the one considered by Y. Zhang for weighted group algebras. Our methods can be applied to various families of weighted Orlicz algebras, including weighted $L^p$-spaces.
Let $G$ be a locally compact group, and let $Φ$ be a Young function. In this paper, we give sufficient and necessary conditions for weighted translation operators on the Orlicz space $L^Φ(G)$ to be chaotic and topologically multiply recurrent. In particular, chaos implies multiple recurrence in our case.
In this paper we present sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces which complete the Osançliol result in 2014. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in $\mathbb{R}^n$.
In this note we consider weighted conditional type operators between different Orlicz spaces and generalized conditional type Holder inequality that we defined in [2]. Then we give some necessary and sufficient conditions for boundedness of weighted conditional type operators. As a consequence we characterize boundedness of weighted conditional type operators and multiplication operators between different L^p-spaces. Finally, we give some upper and lower bounds for essential norm of weighted conditional type operators.
Let $G$ be a locally compact group, and let $Φ$ be a Young function. In this paper, we give a sufficient and necessary condition for weighted translations on the Orlicz space $L^Φ(G)$ to be disjoint topologically transitive. This characterization for disjoint chaos follows from the investigation on disjoint transitivity immediately.
In this paper, we obtain the necessary and sufficient conditions for the weak/strong boundedness of the Calderón-Zygmund operators in generalized weighted Orlicz-Morrey spaces. We also study the boundedness of the commutators of Calderón-Zygmund operators on these spaces. Moreover, the boundedness of Calderón-Zygmund operators in the vector-valued setting is 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 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.
We prove asymptotic formulas for Toeplitz determinants generated by functions with sequences of Fourier coefficients belonging to weighted Orlicz sequence classes. We concentrate our attention on the case of nonvanishing generating functions with nonzero Cauchy index.
In the present paper we prove Besocovich weighted ergodic theorem for positive contractions acting on Orlich-Kantorovich space. Our main tool is the use of methods of measurable bundles of Banach-Kantorovich lattices.
Let $Φ_1$ and $Φ_2$ be nondecreasing functions from $\mathbb{R_+}=(0,\infty)$ onto itself. For $i=1,2$ and $γ\in \mathbb{R}$, define the Orlicz class $L_{Φ_{i}}(\mathbb{R_+})$ to be the set of Lebesgue-measurable functions $f$ on $\mathbb{R_+}$ such that \begin{equation*} \int_{\mathbb{R_+}} Φ_{i} \left( k|(Tf)(t)| \right) t^γdt < \infty \end{equation*} for some $k>0$. Our goal in this paper is to find conditions on $Φ_1$, $Φ_2$, $γ$ and an operator $T$ so that the assertions \begin{equation} T : L_{Φ_2,t^γ}(\mathbb{R_+}) \rightarrow L_{Φ_1,t^γ}(\mathbb{R_+}), \tag{I} \end{equation} and \begin…
We give embedding theorems for weighted Bergman-Orlicz spaces on the ball and then apply our results to the study of composition operators in this context. As one of the motivations of this work, we show that there exist some weighted Bergman-Orlicz spaces, different from $H^{\infinity}$, on which every composition operator is bounded.
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.
We show that the weighted Bergman-Orlicz space $A\_α^ψ$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $ψ$ satisfies the so-called $Δ^{2}$--condition. In addition we prove that this condition characterizes those $A\_α^ψ$ on which every composition operator is bounded or order bounded into the Orlicz space $L\_α^ψ$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $ψ$ satisfies the $Δ^{2}$--condition, a composition operator is compact on $A\_α^ψ$ if and only …
In this paper, we characterize hypercyclic sequences of weighted translation operators on an Orlicz space in the context of locally compact hypergroups.
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）
Let $(Ω,\mathcal{F},\mathbb{P})$ be a probability space and $φ:\ Ω\times[0,\infty)\to[0,\infty)$ be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space $L^φ(Ω)$. Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on $L^φ(Ω)$. As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for $L^φ(Ω)$, which are new even in weighted Orlicz spaces. The authors then establish the atomic characterization…
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…
The best Musielak-Orlicz approximation by linear subspaces
Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS), CONICET–UBA（路易斯·A·桑塔洛数学研究所（IMAS），CONICET-布宜诺斯艾利斯大学）；Instituto de Matemática Aplicada San Luis Dr. Ezio Marchi (IMASL), CONICET–UNSL（圣路易斯埃齐奥·马尔基应用数学研究所（IMASL），CONICET-圣路易斯国立大学）；Departamento de Matemática, FCEFQyN, Universidad Nacional de Río Cuarto – CONICET（里奥夸尔托国立大学理学院、科学学院和自然科学系数学系，CONICET）
本文研究 Musielak-Orlicz 空间中的最佳模逼近，通过 Orlicz 函数原点附近的几何分类，在最小假设下建立存在性结果，并利用单侧导数刻画极小元，最终给出唯一性准则，包括 Haar 型定理和高维空间中的刻画。
Let $(Ω,\mathcal{F},\mathbb{P})$ be a probability space and $φ:\ Ω\times[0,\infty)\to [0,\infty)$ be a Musielak--Orlicz function. In this article, the authors establish the atomic characterizations of weak martingale Musielak--Orlicz Hardy spaces $WH_φ^s(Ω)$, $WH_φ^M(Ω)$, $WH_φ^S(Ω)$, $WP_φ(Ω)$ and $WQ_φ(Ω)$. Using these atomic characterizations, the authors then obtain the boundedness of sublinear operators from weak martingale Musielak--Orlicz Hardy spaces to weak Musielak--Orlicz spaces, and some martingale inequalities which further clarify the relationships among $WH_φ^s(Ω)$, $WH_φ^M(Ω)$,…
Let $Φ$ be a concave function on $(0,\infty)$ of strictly lower type $p_Φ\in(0,1]$ and $ω\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$. We introduce the weighted local Orlicz-Hardy space $h^Φ_ω(\mathbb{R}^n)$ via the local grand maximal function. Let $ρ(t)\equiv t^{-1}/Φ^{-1}(t^{-1})$ for all $t\in(0,\infty)$. We also introduce the $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_{ρ,\,ω}(\mathbb{R}^n)$ and establish the duality between $h^Φ_ω(\mathbb{R}^n)$ and $\mathop\mathrm{bmo}_{ρ,\,ω}(\mathbb{R}^n)$. Several real-varaiable characterizations of $h^Φ_ω(\mathbb{R}^n)$ are presente…
We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets; unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum, that is, by a subset of the group; the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2; the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.
In this article we investigate the Fourier series and transforms for the functions defined on the $ [0, 2 π]^ d $ or $ R^d $ and belonging to the exponential Orlicz and some other rearrangement invariant (r.i.) spaces.
Some new examples of K-monotone couples of the type (X, X(w)), where X is a symmetric space on [0, 1] and w is a weight on [0, 1], are presented. Based on the property of the w-decomposability of a symmetric space we show that, if a weight w changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X, X(w)) is K-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of X is t^{1/p} for some p \in [1, \infty], then X = L_p. At the same time a Banach couple (X, X(w)) may be K-monotone for some non…
The main goal of this paper is to prove a two-weight criteria for multidimensio-nal Hardy type operator from weighted Lebesgue spaces into $p$-convex weighted Banach function spaces. Analogously problem for the dual operator is considered. As application we prove a two-weight criteria for boundedness of multidimensional geometric mean operator and sufficient condition on the weights for boundedness of certain sublinear operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.
We consider multiplication properties of elements in weighted Fourier Lebesgue and modulation spaces. Especially we extend some results by Pilipovic, Teofanov and Toft (2010).
Let $φ$ be a Musielak-Orlicz function satisfying that, for any $(x,\,t)\in\mathbb{R}^n\times(0,\,\infty)$, $φ(\cdot,\,t)$ belongs to the Muckenhoupt weight class $A_\infty (\mathbb{R}^n)$ with the critical weight exponent $q(φ)\in[1,\,\infty)$ and $φ(x,\,\cdot)$ is an Orlicz function with $0<i(φ)\le I(φ)\le 1$ which are, respectively, its critical lower type and upper type. In this article, the authors establish the Riesz transform characterizations of the Musielak-Orlicz-Hardy spaces $H_φ(\mathbb{R}^n)$ which are generalizations of weighted Hardy spaces and Orlicz-Hardy spaces. Precisely, the…
Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential operators.
The strong dual space of linear continuous functionals on a weighted space G of infinitely differentiable functions defined on the real line is described in terms of their Fourier-Laplace transforms.
Let $φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight. The Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$ is defined to be the space of all $f\in{\mathcal S}'({\mathbb R^n})$ such that the grand maximal function $f^*$ belongs to the Musielak-Orlicz space $L^φ(\mathbb R^n)$. Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of $H^φ(\mathbb R^n)$ in terms of the vertical or the non-tangential maximal f…
