We prove that the Hardy-Littlewood maximal operator is bounded in the weighted generalized Orlicz space if the weight satisfies the classical Muckenhoupt condition $A_p$ and $t \to \frac{φ(x,t)}{t^p}$ is almost increasing in addition to the standard conditions.
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.
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.
Boundedness of the maximal function and the Caldeón-Zygmund singular integrals in central Morrey-Orlicz spaces were proved in papers by the second and third authors. The weak-type estimates have also been proven. Here we show boundedness of the Riesz potential in central Morrey-Orlicz spaces and the corresponding weak-type version.
In the present paper, we shall give a characterization for weak/strong Adams-type boundedness of the fractional maximal operator on generalized Orlicz-Morrey spaces.
We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces. To do this we prove the weak-weak type modular inequality of the Hardy-Littlewood maximal operator with respect to the Young function. Orlicz-Morrey spaces contain $L^p$ spaces ($1\le p\le\infty$), Orlicz spaces and generalized Morrey spaces as special cases. Hence we get necessary and sufficient conditions on these function spaces as corollaries.
In the present paper, we will characterize the boundedness of the generalized fractional integral operators $I_ρ$ and the generalized fractional maximal operators $M_ρ$ on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces.
In this paper, we investigate the boundedness of maximal operator and its commutators in generalized Orlicz-Morrey spaces on the spaces of homogeneous type. As an application of this boundedness, we give necessary and sufficient condition for the Adams type boundedness of fractional integral and its commutators in these spaces. We also discuss criteria for the boundedness of these operators in Orlicz spaces.
Boundedness of composition operator in Orlicz-Morrey spaces
Graduate School of Information Science and Technology, The University of Osaka（大阪大学大学院信息科学技术研究科）；Center for Advanced Intelligence Project RIKEN（理化学研究所先进智能项目中心）；Center for Science Adventure and Collaborative Research Advancement (SACRA), Graduate School of Science, Kyoto University（京都大学理学研究院科学冒险与合作研究推进中心）；Center for Mathematics and Data Science, Gunma University（群马大学数学与数据科学中心）
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.
We study sharp growth conditions for the boundedness of the Hardy-Littlewood maximal function in the generalized Orlicz spaces. We assume that the generalized Orlicz function $ϕ(x, t)$ satisfies the standard continuity properties (A0), (A1) and (A2). We show that if the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space to itself then $ϕ(x,t)/ t^p$ is almost increasing for large $t$ for some $p>1$. Moreover we show that the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space $L^ϕ(\mathbb{R}^n)$ to itself if and only if $ϕ$ is weakly equiva…
A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into $L^\infty(\mathbb R^n)$ fails. They also augment a classical embedding t…
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.
In the article we generalize the Marcinkiewicz sampling theorem in the context of Orlicz spaces. We establish conditions under which sampling theorem holds in terms of restricted submultiplicativity and supermultiplicativity of an $N$-function $φ$, boundedness of the Hilbert transform and Matuszewska-Orlicz indices. In addition we give a new criterion for boundedness of Hilbert transform on Orlicz space.
We investigate the commutators $[b,I_ρ]$ of generalized fractional integral operators $I_ρ$ with functions $b$ in generalized Campanato spaces and give a necessary and sufficient condition for the boundedness of the commutators on Orlicz spaces. To do this we define Orlicz spaces with generalized Young functions and prove the boundedness of generalized fractional maximal operators on the Orlicz spaces.
In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation. To best of our knowledge, these results are also new in the special case of double phase spaces.
We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\mathbb{R}^{n})$ including their weak versions $WM_{Φ,φ}(\mathbb{R}^{n})$. We find the sufficient conditions on the pairs $(φ_{1},φ_{2})$ and $(Φ, Ψ)$ which ensures the boundedness of the fractional maximal operator $M_α$ from $M_{Φ,φ_1}(\mathbb{R}^{n})$ to $M_{Ψ,φ_2}(\mathbb{R}^{n})$ and from $M_{Φ,φ_1}(\mathbb{R}^{n})$ to $WM_{Ψ,φ_2}(\mathbb{R}^{n})$. As applications of those results, the boundedness of the commutators of the fractional maximal operator $M_{b,α}$ with $b \in BMO(\mathbb{R}^{n})$ on the spaces $M_{Φ,φ}(\mathbb{R}^{n})$ is …
We consider the commutators $[b,T]$ and $[b,I_ρ]$ on Orlicz-Morrey spaces, where $T$ is a Calderón-Zygmund operator, $I_ρ$ is a generalized fractional integral operator and $b$ is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness of the commutators on Orlicz-Morrey spaces. To do this we prove the boundedness of generalized fractional maximal operators on Orlicz-Morrey spaces. Moreover, we introduce Orlicz-Campanato spaces and establish their relations to Orlicz-Morrey spaces.
In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator $M_α$ on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator $M_{b,α}$ and nonlinear commutator of fractional maximal operator $[b,M_α]$ on Orlicz spaces, when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.
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.
We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\Rn)$ including their weak versions $WM_{Φ,φ}(\Rn)$. In these spaces we prove the boundedness of the Riesz potential from $M_{Φ,φ_1}(\Rn)$ to $M_{Ψ,φ_2}(\Rn)$ and from $M_{Φ,φ_1}(\Rn)$ to $WM_{Ψ,φ_2}(\Rn)$. As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(φ_{1},φ_{2})$, which do not assume any assumption on monotonicity of $φ_{…
In this paper we study some basic properties, like boundedness and closedness of range, of multiplication conditional expectation(MCE) operators between different Orlicz spaces.
In this paper, we consider the predual spaces of weak Orlicz spaces. As an application, we provide the Fefferman-Stein vector-valued maximal inequality for the weak Orlicz spaces. In order to prove this statement, we introduced the Orlicz-Lorentz spaces, and showed the boundedness of the Hardy-Littlewood maximal operator on these spaces.
We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,φ}(\mathbb{R}^n)$. Under some natural assumptions on $φ$ we show that the maximal function is bounded and continuous in $W^{1,φ}(\mathbb{R}^n)$.
We study the fractional maximal operator acting between Orlicz spaces. We characterise whether the operator is bounded between two given Orlicz spaces. Also a necessary and sufficient conditions for the existence of an optimal target and domain spaces are established and the explicit formulas of corresponding optimal Young functions are presented.
In this paper we consider composition operator $C_φ generated by nonsingular measurable transformation $T$ and multiplication operator $M_u$ generated by measurable function $u$ between two different Orlicz spaces, then we investigate boundedness, compactness and essential norm of multiplication and composition operators in term of properties of the mapping $φ$, the function $u$ and the measure space $(X, Σ, μ)$.
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.
We give necessary and sufficient conditions for the boundedness of the maximal commutators $M_{b}$, the commutators of the maximal operator $[b, M]$ and the commutators of the sharp maximal operator $[b, M^{\sharp}]$ in Orlicz spaces $L^Φ(\mathbb{G})$ on any stratified Lie group $\mathbb{G}$ when $b$ belongs to Lipschitz spaces $\dotΛ_β(\mathbb{G})$. We obtain some new characterizations for certain subclasses of Lipschitz spaces $\dotΛ_β(\mathbb{G})$.
Let $φ: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\vz(x,\cdot)$ is nondecreasing, $φ(x,0)=0$, $φ(x,t)>0$ when $t>0$, $\lim_{t\to\infty}φ(x,t)=\infty$ and $\vz(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight uniformly in $t$. Let $ϕ: [0,\infty)\to[0,\infty)$ be nondecreasing. In this article, the authors introduce the Musielak-Orlicz Morrey space $\mathcal M^{φ,ϕ}(\mathbb R^n)$ and obtain the boundedness on $\mathcal M^{φ,ϕ}(\mathbb R^n)$ of the intrinsic Lusin area function $S_α$, the intrinsic $g$-function $g_α$, the intrinsic $g_λ^*$-function $g^\ast_{λ, α}$ and …
We introduce a new class of Hardy spaces $H^{φ(\cdot,\cdot)}(\mathbb R^n)$, called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of García-Cuerva, Strömberg, and Torchinsky. Here, $φ: \mathbb R^n\times [0,\infty)\to [0,\infty)$ is a function such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight. A function $f$ belongs to $H^{φ(\cdot,\cdot)}(\mathbb R^n)$ if and only if its maximal function $f^*$ is so that $x\mapsto φ(x,|f^*(x)|)$ is integrable. Such a space arises naturally for …
In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson in…
We characterize the relatively compact subsets of $L^1\left(\| m \| \right),$ the quasi-Banach function space associated to the semivariation of a given vector measure $m$ showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classic setting of Lebesgue spaces remains almost invariant in this new context of the Choquet integration. Also we present a de la Vallée-Poussin type theorem in the context of these spaces $L^1\left(\|m\|\right)$ that allows us to locate each compact…
By the Hardy-Littlewood-Sobolev theorem the classical Riesz potential is bounded on Lebesgue spaces. E. Nakai and H. Sumitomo [16] extended that theorem to the Orlicz spaces. We introduce generalized potential operators on commutative hypergroups and under some assumptions on the kernel we showed the boundedness of these operators from Lebesgue space into certain Orlicz space. Our result is an analogue of Theorem 1.3 in [16].
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 $WH_φ^s(Ω)$, $WH_φ^M(Ω)$,…
In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This pap…
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
In this paper, we prove strong type, weak type inequalities of Hardy-Littlewood maximal operator and fractional Hardy-Littlewood maximal operator on variable sequence spaces lp(Z). This is achieved using Calderon-Zygmund decomposition for sequences, properties of modular functional and Log Holder continuity.
Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.
We investigate Sobolev spaces $W^{1,Φ}$ associated to Musielak-Orlicz spaces $L^Φ$. We first present conditions for the boundedness of the Voltera operator in $L^Φ$. Employing this, we provide necessary and sufficient conditions for $W^{1,Φ}$ to contain isomorphic subspaces to $\ell^\infty$ or $\ell^1$. Further we give necessary and sufficient conditions in terms of the function $Φ$ or its complementary function $Φ^*$ for reflexivity, uniform convexity, $B$-convexity and superreflexivity of $W^{1,Φ}$. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,φ}$ where …
In this paper we prove that maximal H-monotone operators $T:H^n\rightrightarrows V_1$ whose domain is all the Heisenberg group $H^n$ are locally bounded. This implies that they are upper semicontinuous. As a consequence, maximal H-monotonicity of an operator on $H^n$ can be characterized by a suitable version of Minty's type theorem.
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…
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.
Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of several different commutators of bilinear Hardy-Littlewood maximal function, respectively; these commutators include the maximal iterated commutator $\mathcal{M}_{Π\vec{b}}$, the maximal linear commutator $\mathcal{M}_{Σ\vec{b}}$, the iterated commutator $[Π\vec{b},\mathcal{M}]$ and the linear commutator $[Σ\vec{b},\mathcal{M}]$.
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a $p$-Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among others, that a suitable maximal operator is locally weakly bounded. In particular, various suff…
In this paper, we first give some new characterizations of Muckenhoupt type weights through establishing the boundedness of maximal operators on the weighted Lorentz and Morrey spaces. Secondly, we establish the boundedness of sublinear operators including many interesting in harmonic analysis and its commutators on the weighted Morrey spaces. Finally, as an application, the boundedness of strongly singular integral operators and commutators with symbols in BMO space are also given.
We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of t…
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…
In this paper, we show the strong and weak type boundedness of $T_{Ω,α}^A$ and $M_{Ω,α}^A$, the multilinear fractional integral operators and the corresponding fractional maximal operators, on the two weights weighted Morrey space. Furthermore, we also obtain the mulitilinear singular integral operators $T_Ω^A$ and the corresponding maximal operators $M_Ω^A$ are strong bounded operators on the same spaces.
