Fahreddin Abdullayev, Stanislav Chaichenko, Andrii Shidlich
In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.
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Let $φ:\mathbb{R}^n\times[0,\,\infty) \rightarrow [0,\,\infty)$ satisfy that $φ(x,\,\cdot)$, for any given $x\in\mathbb{R}^n$, is an Orlicz function and $φ(\cdot\,,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t\in(0,\,\infty)$. The Musielak-Orlicz Hardy space $H^φ(\mathbb{R}^n)$ generalizes both of the weighted Hardy space and the Orlicz Hardy space and hence has a wide generality. In this paper, the authors first prove the completeness of both of the Musielak-Orlicz space $L^φ(\mathbb{R}^n)$ and the weak Musielak-Orlicz space $WL^φ(\mathbb{R}^n)$. Then the authors obtain two boundedne…
In this paper, the author establishes the boundedness of parametric Littlewood-Paley operators from Musielak-Orlicz Hardy space to Musielak-Orlicz space, or to weak Musielak-Orlicz space at the critical index. Part of these results are new even for classical Hardy space of Fefferman and Stein.
Characterizations of a class of Musielak--Orlicz BMO spaces via commutators of Riesz potential operators
本文研究了通过Riesz势算子的交换子将Musielak-Orlicz Hardy空间映射到Musielak-Orlicz空间的条件，提出了新的BMO子空间定义并验证了其有界性。
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 $WH_φ^s(Ω)$, $WH_φ^M(Ω)$,…
Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications
Musielak-Orlicz Hardy空间的Lusin面积函数和分子特征及其应用设$φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$是一个增长函数，使得$φ(x,\cdot)$是非减的，当$t>0$，$\lim_{t\to\infty}φ(x,t)=\infty$，$φ(\cdot,t)$是一个Muckenhoupt $A_\infty(\mathbb{R}^n)$在$t$中一致加权时，$φ(x,0)=0$，$φ(x,t)>0$.利用极大函数建立了Luong Dang Ky引入的Musielak-Orlicz Hardy空间$H_φ(\mathbb{R}^n)$的Lusin面积函数和分子特征.作为应用，我们得到了Musielak-Orlicz ${\mathop\mathrm{BMO}}$型空间$\mathop\mathrm{BMO}_φ(\mathbb{R}^n)$的$φ$-Carleson测度刻画，该空间已被Luong Dang Ky证明是$H_φ(\mathbb{R}^n)$的对偶空间.
New Real-Variable Characterizations of Musielak-Orlicz Hardy Spaces
令$φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$使得$φ(x,\cdot)$是Orlicz函数且$φ(\cdot,t)$是Muckenhoupt $A_\infty({\mathbb R^n})$权重。Musielak-Orlicz Hardy空间$H^φ(\mathbb R^n)$被定义为所有$f\in{\mathcal S}'({\mathbb R^n})$的空间，使得极大函数$f^*$属于Musielak-Orlicz空间$L^φ(\mathbb R^n)$。梁当基建立了它的原子特征。本文通过建立Musielak-Orlicz Festhiman-Stein向量值不等式，利用垂直或非切向极大函数，或Littlewood-Paley $g$ -函数或$g_λ^\ast$ -函数，建立了$H^φ(\mathbb R^n)$的一些新的实变量刻画.此外，当$H^φ(\mathbb R^n)$是经典Hardy空间$H^p(\mathbb R^n)$，具有$p\in (0,1]$或其加权变体时，$H^φ(\mathbb R^n)$的$g_λ^\ast$函数表征中的$λ$的范围与已知的最佳结果一致。
Musielak-Orlicz Campanato Spaces and Applications
设$φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$是这样的，$φ(x,\cdot)$是Orlicz函数，$φ(\cdot,t)$是Muckenhoupt $A_\infty(\mathbb R^n)$在$t$中均匀加权。本文引入了Musielak-Orlicz Campanato空间$H^φ(\mathbb R^n)$，作为应用，证明了在L. D. Ky [arXiv：1105.0486].建立了${\mathcal L}_{φ,1,s}({\mathbb R}^n)$中函数的John-Nirenberg不等式，作为应用，得到了${\mathcal L}_{φ,q,s}({\mathbb R}^n)$的几个等价刻画，进而导出了${\mathcal L}_{φ,1,s}({\mathbb R}^n)$的$φ$-Carleson测度刻画.
In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal S}_{\bf M}$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.
Littlewood-Paley Characterization for Musielak-Orlicz-Hardy Spaces Associated with Operators
本文基于空间类型X，研究了与非负二阶自伴随算子L相关的穆西尔克-奥里奇-哈代空间H_{φ,L}(X)的利特伍德-帕莱伊函数特征，证明了其与H_{L,G,φ}(X)的等价性。
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 $f\in L_{\ma…
New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators
引入一类新的Hardy空间$H^{φ(\cdot,\cdot)}(\mathbb R^n)$，称为Musielak-Orlicz型Hardy空间，它推广了Janson的Hardy-Orlicz空间和García-Cuerva，Strömberg，Torchinsky的加权Hardy空间.这里，$φ: \mathbb R^n\times [0,\infty)\to [0,\infty)$是使得$φ(x,\cdot)$是Orlicz函数并且$φ(\cdot,t)$是Muckenhoupt $A_\infty$权重的函数。一个函数$f$属于$H^{φ(\cdot,\cdot)}(\mathbb R^n)$当且仅当它的极大函数$f^*$使得$x\mapsto φ(x,|f^*(x)|)$是可积的。这样的空间例如在分别描述$H^1(\mathbb R^n)$和$BMO(\mathbb R^n)$中的函数的乘积时自然出现（参见\cite{BGK}）。我们通过极大函数刻画了这些空间，并建立了它们的原子分解。我们还刻画了它们的对偶空间。由Nakai和Yabuta刻画的$BMO(\mathbb R^n)$的点态乘子类可以看作是$L^1(\mathbb R^n)+ H^{\rm log}(\mathbb R^n)$的对偶，其中$ H^{\rm log}(\mathbb R^n)$是与Musielak-…
Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $φ:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the spac…
Musielak-Orlicz Hardy Spaces Associated with Operators and Their Applications
设$\mathcal{X}$是具有加倍测度的度量空间，$L$是$L^2(\mathcal{X})$中满足Davies-Gaffney估计的非负自伴算子.设$φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$是一个函数，使得$φ(x,\cdot)$是一个Orlicz函数，$φ(\cdot,t)\in A_{\infty}(\mathcal{X})$（Muckenhoupt权类）及其一致临界下型指标$i(φ)\in(0,1]$。本文通过与$L$生成的热半群相联系的Lusin面积函数引入了Musielak-Orlicz Hardy空间$H_{φ,\,L}(\mathcal{X})$，并进一步证明了Musielak-Orlicz $\mathop\mathrm{BMO}$型空间$\mathop\mathrm{BMO}_{φ,\,L}(\mathcal{X})$是$H_{φ,\,L}(\mathcal{X})$的对偶空间，作为推论，得到了$\mathop\mathrm{BMO}_{φ,\,L}(\mathcal{X})$的$φ$-Carleson测度刻画.给出了$H_{φ,\,L}(\mathcal{X})$的特征，包括原子、分子和与$L$的Poisson半群相联系的Lusin面积函数.利用原子表征方法，作者用$g^\ast_{λ,\,L}$表征了$H…
Musielak-Orlicz BMO-Type Spaces Associated with Generalized Approximations to the Identity
设$\mathcal{X}$是齐型空间，$φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$是增长函数，使得$φ(\cdot,t)$是$t$中一致的Muckenhoupt权，$φ(x,\cdot)$是一致上1型和下$p\in(0,1]$型的Orlicz函数.本文引入了一个新的Musielak-Orlicz BMO型空间$\mathrm{BMO}^φ_A(\mathcal{X})$，给出了它的基本性质，并分别建立了它关于空间$\mathrm{BMO}^φ_{A,\,\mathrm{max}}(\mathcal{X})$和$\widetilde{\mathrm{BMO}}^φ_A(\mathcal{X})$的两个等价刻画.此外，还得到了$\mathrm{BMO}^φ_A(\mathcal{X})$上John-Nirenberg不等式的两个变形.作为应用，进一步证明了与Laplace算子$Δ$在$\mathbb{R}^n$上的Poisson半群相联系的空间$\mathrm{BMO}^φ(\mathbb{R}^n)$与L. D.肯塔基州
Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates
设$\mathcal{X}$是具有加倍测度的度量空间，$L$是$ω$型的一对一算子，在$L^2(\mathcal{X})$中有界$H_\infty$ -泛函演算满足球上的加强$(p_L, q_L)$非对角估计，其中$p_L\in[1,2)$和$q_L\in(2,\infty]$ .设$φ:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$是一个函数，使得$φ(x,\cdot)$是一个Orlicz函数，$φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$（一致Muckenhoupt权类），它的一致临界上型指标$I(φ)\in(0,1]$和$φ(\cdot,t)$满足$(q_L/I(φ))'$阶的一致逆Hölder不等式.通过与$L$相联系的Lusin面积函数，引入了一个Musielak-Orlicz-Hardy空间$H_{φ,\,L}(\mathcal{X})$，并建立了它的分子特征.特别地，当$L$是非负自伴的且满足Davies-Gaffney估计时，还得到了$H_{φ,\,L}(\mathcal{X})$的原子特征.进一步给出了$H_{φ,\,L}(\mathbb{R}^n)$与经典Musielak-Orlicz-Hardy空间$H_φ(\mathbb{R}^n)$等价的一个充分条件.此外，对于$\…
Local Hardy Spaces of Musielak-Orlicz Type and Their Applications
令$ϕ: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz)$是一个函数，使得$ϕ(x,\cdot)$是Orlicz函数，并且$ϕ(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$（由VS. Rychkov）。利用局部极大函数引入了Musielak-Orlicz型局部Hardy空间$h_ϕ(\mathbb{R}^n)$，并进一步证明了局部$\mathop\mathrm{BMO}$型空间$\mathop\mathrm{bmo}_ϕ(\mathbb{R}^n)$是$h_ϕ(\mathbb{R}^n)$的对偶空间.作为应用，证明了E. Nakai和K. Yabuta是$L^1(\rn)+h_{Φ_0}(\mathbb{R}^n)$的对偶，其中$ϕ$是满足某些附加生长条件的$(0,\infty)$上的增函数，而$Φ_0$是由$ϕ$诱导的Musielak-Orlicz函数。给出了$h_ϕ(\mathbb{R}^n)$的特征，包括原子、局部垂直极大函数和局部非切极大函数.利用原子刻画，证明了在$h_ϕ(\mathbb{R}^n)$的某些稠密子空间中存在有限的达到范数的原子分解，并由此进一步导出了某些次线性算子在$h_ϕ(\mathbb{R}^n)$上有界的判别准则.最后证明了局部Riesz…
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…
Boundedness of Intrinsic Littlewood-Paley Functions on Musielak-Orlicz Morrey and Campanato Spaces
设$φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$为$\vz(x,\cdot)$非减的，当$t>0$、$\lim_{t\to\infty}φ(x,t)=\infty$和$\vz(\cdot,t)$是在$t$中均匀加权的Muckenhoupt $A_\infty({\mathbb R^n})$时，$φ(x,0)=0$，$φ(x,t)>0$。设$ϕ: [0,\infty)\to[0,\infty)$是非递减的。引入Musielak-Orlicz Morrey空间$\mathcal M^{φ,ϕ}(\mathbb R^n)$，得到了固有Lusin面积函数$S_α$，固有$g$ -函数$g_α$，固有$g_λ^*$-函数$g^\ast_{λ, α}$及其与${\rm BMO}(\rn)$函数的交换子在$\mathcal M^{φ,ϕ}(\mathbb R^n)$上的有界性，其中$α\in(0,1]$，$λ\in(\min\{\max\{3,\,p_1\},3+2\az/n\},\infty)$和$p_1$表示$\vz$的一致上型指数.设$Φ: [0,\infty)\to[0,\infty)$不减，$Φ(0)=0$，$Φ(t)>0$当$t>0$，和$\lim_{t\to\infty}Φ(t)=\infty$，$w\in A_\infty(\ma…
For any $p\in(0,\,1]$, let $H^{Φ_p}(\mathbb{R}^n)$ be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function $Φ_p$, defined by setting, for any $x\in\mathbb{R}^n$ and $t\in[0,\,\infty)$, $$ Φ_{p}(x,\,t):= \begin{cases} \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}} & \qquad \text{when } n(1/p-1)\notin \mathbb{N} \cup \{0\}; \\ \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}[\log(e+|x|)]^p} & \qquad \text{when } n(1/p-1)\in \mathbb{N}\cup\{0\},\\ \end{cases} $$ which is the sharp target space of the bilinear decomposition of the product of the Hardy space $H^p(\mathbb{R}^n)$ and its …
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characteri…
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…
Real-variable Theory of Anisotropic Musielak-Orlicz-Lorentz Hardy Spaces with Applications to Calderón-Zygmund Operators
本文引入各向异性 Musielak-Orlicz-Lorentz Hardy 空间，建立其原子与分子刻画，并证明 Calderón-Zygmund 算子的有界性，将已知结果从各向同性推广至各向异性并放宽指数范围。
BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces
本文研究内蕴平方函数交换子，证明其有界性等价于BMO型空间成员资格，并将加权结果推广至Musielak-Orlicz与Orlicz-Hardy空间，给出端点估计。
In the Orlicz type spaces ${\mathcal S}_{M}$, we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre $K$-functionals in the spaces ${\mathcal S}_{M}$.
Direct and inverse approximation theorems in the Besicovitch-Museilak-Orlicz spaces of almost periodic functions
本文在 Besicovitch-Musielak-Orlicz 空间中，借助最佳逼近和广义光滑模，为一类 Fourier 指数受限的殆周期函数建立正、逆逼近定理，并考察常数的不可改进性。
In weighted Orlicz type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,μ}}$ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is in a certain sense the best. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre …
We prove that a generalized Fefferman-Phong type condition on a pair of weights $u$ and $v$ is sufficient for the boundedness of the commutators of potential type operators from $L^{p(\cdot)}_v$ into $L^{q(\cdot)}_u$. We also give an improvement of this result in the sense that we not only consider a variable version of power bump conditions, but also weaker norms related to Musielak-Orlicz functions. We consider a wider class of symbols including Lipschitz symbols and some generalizations.
In this paper we obtained some direct and inverse theorems of approximation theory for $ψ$-differentiable functions in the metric weighted Orlicz spaces with weights, which belong to the class of Muckenhoupt.
This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log}$ and related Musielak-Orlicz spaces.
We give an atomic decomposition of closed forms on R n , the coefficients of which belong to some Hardy space of Musielak-Orlicz type. These spaces are natural generalizations of weighted Hardy-Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called H log , appears naturally when considering products of functions in the Hardy space H 1 and in BM O. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in H log. Namely, a closed form in H log is the infinite sum of the wedge product between an …
In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt's $A_{p}$ condition. To obtain refined version of the Jackson type inequality we prove an extrapolation theorem, Marcinkiewicz multiplier theorem and Littlewood Paley type results. As a consequence refined inverse Marchaud type inequalities are obtained. By means of a realization result we find an equivalence between the fractional order weighted modulus of smoothness and the classical weighted Peetre's K-functional.
Convexity of the embedding parameter sets of some analytic function spaces
研究加权Bergman-Orlicz空间连续嵌入参数集的几何结构，证明在特定对数凸性和对数凹性条件下，可接受的权重指数集为凸集，并探讨其对偶问题。
Product Hardy spaces meet ball quasi-Banach function spaces
本文通过 Littlewood-Paley-Stein 理论引入了与球拟 Banach 函数空间关联的乘积 Hardy 空间，建立了基于离散 Calderón 恒等式的分解定理，并利用 Rubio de Francia 外推定理证明了双参数奇异积分算子的有界性。
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