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.
In this paper, we establish multilinear BMO estimates for commutators of multilinear fractional maximal and integral operators both on product generalized Morrey spaces and product generalized vanishing Morrey spaces, respectively. Similar results are still valid for commutators of multilinear maximal and singular integral operators.
The aim of this paper is to prove the boundedness of the oscillation and variation operators for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces.
The purpose of this paper is to establish some neccessary and sufficient conditions for the boundedness of a general class of multilinear Hausdorff operators that acts on the product of some two weighted function spaces such as the two weighted Morrey, Herz and Morrey-Herz spaces. Moreover, some sufficient conditions for the boundedness of multilinear Hausdorff operators on the such spaces with respect to the Muckenhoupt weights are also given.
In this paper, we establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of p-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.
The aim of this paper is to introduce and study the boundedness of a new class of p-adic rough multilinear Hausdorff operators on the product of Herz, central Morrey and Morrey-Herz spaces with power weights and Muckenhoupt weights. We also establish the boundedness for the commutators of p-adic rough multilinear Hausdorff operators on the weighted spaces with symbols in central BMO space.
In this paper, we consider the boundedness of fractional type multilinear commutators generated by fractional integral with rough variable kernel and local Campanato functions on both generalized local (central) Morrey spaces and generalized vanishing local Morrey spaces, under generic size conditions which are satisfied by most of the operators in harmonic analysis, respectively.
In this paper, we give the boundedness of some parabolic multilinear commutators generated by a class of parabolic maximal and linear operators with rough kernel and parabolic local Campanato functions on the parabolic generalized local Morrey spaces, respectively. Indeed, the results in this paper are extensions of some known results.
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.
Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(\mathbb{R}^n)$ with Gaussian kernel bounds, and let $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n$. Assume that $\vec{b}=(b_1,b_2,\cdots,b_m)$ is a finite family of locally integrable functions, then the multilinear commutators generated by $\vec{b}$ and $L^{-α/2}$ is defined by \begin{equation*} L_{\vec{b}}^{-α/2}f=[b_m,\cdots,[b_2,[b_1,L^{-α/2}]],\cdots,]f \end{equation*} when $b_j\in BMO(w)$, $j=1,2,\cdots,m$, the authors obtain the boundedness of $L_{\vec{b}}^{-α/2}$ on weighted Morrey spaces.
In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and …
The bidual of the closure of smooth functions with respect to the Morrey norm coincides with the Morrey space. This assertion is generalized to some Muckenhoupt weighted Morrey spaces. We combine this fact with basic extrapolation techniques due to Rubio de Francia adapted to weighted Morrey spaces. This leads to new results on the boundedness of operators even for the unweighted case.
In this article, we establish some conditions for the boundedness of fractional integral operators on the vanishing generalized weighted Morrey spaces. We also investigate corresponding commutators generated by BMO functions.
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 this paper, we prove the Spanne-type boundedness of the generalized Riesz potential operator from the one generalized weighted local Morrey spaces to the another one, and from the generalized weighted local Morrey spaces to the weak generalized weighted local Morrey spaces. We also prove the Adams-type boundedness of the operator from the weighted spaces to the another one, and from the weighted spaces to the weak weighted spaces.
In this note we introduce grand grand Morrey spaces, in the spirit of the grand Lebesgue spaces. We prove a kind of \textit{reduction lemma} which is applicable to a variety of operators to reduce their boundedness in grand grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund operators in the framework of grand grand 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 shall give a characterization for weak/strong Adams-type boundedness of the fractional maximal operator on generalized Orlicz-Morrey spaces.
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.
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）
It is proved that both oscillatory integral operators and fractional oscillatory integral operators are bounded on weighted Morrey spaces. The corresponding commutators generated by $BMO$ functions are also considered.
The aim of this paper is to get the boundedness of the commutators of multi-sublinear operators generated by local campanato functions and multilinear Calderón-Zygmund operators on the product generalized local Morrey spaces.
Characterizations for multilinear fractional maximal and integral operators and their commutators on generalized weighted Morrey spaces and applications
This paper includes new bounds concepting the vanishing generalized weighted Morrey space. In this sense, it is outlined improved bounds about the a class of fractional type rough higher order commutators on vanishing generalized weighted Morrey spaces.
In this paper, some boundedness for commutators of fractional integrals are obtained on Herz-Morrey spaces with variable exponent applying some properties of varible exponent and $\BMO$ function.
In this paper, we introduced the local and global mixed Morrey-type spaces, and some properties of these spaces are also studied. After that, the necessary conditions of the boundedness of fractional integral operators $I_α$ are studied respectively in mixed-norm Lebesgue spaces and the local mixed Morrey-type spaces. Last but not least, the boundedness of $I_α$ is obtained by Hardy operators' boundedness in weighted Lebesgue spaces. As corollaries, we acquire the boundedness of $I_α$ in mixed Morrey spaces(\hyperlink{Corollary 5.6}{Corollary 5.6}) and mixed Lebesgue spaces(\hyperlink{Corollar…
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 the boundedness of some sublinear operators on weighted Morrey spaces under certain size conditions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operator, Bochner-Riesz means at the critical index, oscillatory singular operators, singular integral operators with oscillating kernels and so on. As applications, the regularity in weighted Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients are established.
Let $T_m$ be the $m$-th Calderón-Zygmund type singular integral. In the paper, we consider the boundedness of $T_m$ on the generalized product local Morrey spaces $LM_{p_1,φ_1}^{\{x_0\}}\times LM_{p_2,φ_2}^{\{x_0\}}\times\dots\times LM_{p_m,φ_m}^{\{x_0\}}.$ And, the boundedness of the commutators of $T_m$ with local Campanato functions is obtained, also.
In the setting of homogeneous spaces (X,d,μ), it is shown that the commutator of Calderón- Zygmund type operators as well as commutator of potential operator with BMO function are bounded in generalized Grand Morrey space. Interior estimates for solutions of elliptic equations are also given in the framework of generalized grand Morrey spaces.
We study the weighted boundedness of the Cauchy singular integral operator $S_\Gm$ in Morrey spaces $L^{p,λ}(\Gm)$ on curves satisfying the arc-chord condition, for a class of "radial type" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces $L^{p,λ}(0,\ell), \ell>0$. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood max…
In this paper we introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized grand Morrey spaces to the corresponding boundedness in Morrey spaces, as a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator as well as the boundedness of Calderón-Zygmund potential type operators. Boundedness of Riesz type potential operators are also obtained in the …
The aim of this paper is to obtain the boundedness of some operator on grand generalized weighted Morrey spaces $\mathcal{L}^{p),ϕ}_φ(ω)$ over RD-spaces. Under assumption that functions $φ$ and $ϕ$ satisfy certain conditions, the authors prove that Hardy-Littlewood maximal operator and $θ$-type Calderón-Zygmund operator are bounded on grand generalized weighted Morrey spaces $\mathcal{L}^{p),ϕ}_φ(ω)$. Moreover, the boundedness of commutator $[b,T_θ]$ which is generated by $θ$-type Calderón-Zygmund operator $T_θ$ and $b\in\mathrm{BMO}(μ)$ on spaces $\mathcal{L}^{p),ϕ}_φ(ω)$ is also established.…
Dyadic fractional integral operators are shown to be bounded on Morrey spaces and their preduals. It seems that the proof of the boundedness by means of dyadic fractional integral operators is effective particularly on the preduals. In the present paper the commutators are proved to be bounded as well.
In this paper, the authors establish the boundedness for a large class of intrinsic square functions $\mathcal{G}_α$, $g_α$, $g^{\ast}_{\tildeλ,α}$ and their commutators $[b,\mathcal{G}_α]$, $[b,g_α]$ and $[b,g^{\ast}_{\tildeλ,α}]$ generated with $λ$-central $BMO$ functions $b\in CBMO^{p,λ}(\mathbb{R}^{n})$ on generalized central Morrey spaces $\mathcal{B}^{q,φ}(\mathbb{R}^{n})$ for $1<q<\infty,0<α\leq1$, respectively. All of the results are new even on the central Morrey spaces $\mathcal{B}^{q,λ}(\mathbb{R}^{n})$.
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.
We reduce the boundedness of operators in Morrey spaces $L_p^r({\mathbb R}^n)$, its preduals, $H^{\varrho}L_p ({\mathbb R}^n)$, and their preduals $\overset{\circ}{L}{}^r_p({\mathbb R}^n)$ to the boundedness of the appropriate operators in Lebesgue spaces, $L_p({\mathbb R}^n)$. Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications…
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.
We discuss the Hardy-Littlewood maximal operator on discrete Morrey spaces of arbitrary dimension. In particular, we obtain its boundedness on the discrete Morrey spaces using a discrete version of the Fefferman-Stein inequality. As a corollary, we also obtain the boundedness of some Riesz potentials on discrete Morrey spaces.
Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces can not handle. Morrey spaces which we have been handling are called classical Morrey spaces. However, classical Morrey spaces are not totally enough to describe the boundedness properties. To this end, we need to generalize parameters $p$ and $q$, among others $p$.
In this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann-Liouville and Weyl.
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.
The aim of this paper is to get the boundedness of certain sublinear operators with rough kernel generated by Calderón-Zygmund operators on the generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutator operators formed by BMO functions and certain sublinear operators with rough kernel are also bounded on the generalized weighted Morrey spaces. Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.
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.
We prove that for operators satistying weighted inequalities with $A_p$ weights the boundedness on a certain class of Morrey spaces holds with weights of the form $|x|^αw(x)$ for $w\in A_p$. In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N.~Samko for the Hilbert transform, by H.~Tanaka for the Hardy-Littlewood maximal operator, and by S.~Nakamura and Y.~Sawano for Calderón-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Mo…
We define a scale of weighted Morrey spaces which contains different weighted versions appearing in the literature. This allows us to obtain weighted estimates for operators in a unified way. In general, we obtain results for weights of the form $|x|^αw(x)$ with $w\in A_p$ and nonnegative $α$. We study particularly some properties of power-weighted spaces and in the case of the Hardy-Littlewood maximal operator our results for such spaces are sharp. By using extrapolation techniques the results are given in abstract form in such a way that they are automatically obtained for many operators.
In this paper, we introduce homogeneous mixed Herz-Morrey spaces $M\dot{K}_{p,\vec{q}}^{α,λ}(\mathbb{R}^n)$ and show it's some properties. Firstly, the boundedness of sublinear operators, fractional type operators in homogeneous mixed Herz-Morrey spaces is investigated. In particular, the above results are still valid for Calder$\acute{o}$n-Zygmund operators and fractional maximal operators. Lastly, the boundedness of their commutators in homogeneous mixed Herz-Morrey spaces is obtained.
In this paper, we give necessary and sufficient conditions for the boundedness of rough Hausdorff operators on Herz, Morrey and Morrey-Herz spaces with absolutely homogeneous weights. Especially, the estimates for operator norms in each case are worked out. Moreover, we also establish the boundedness of the commutators of rough Hausdorff operators on the such spaces with their symbols belonging to Lipschitz space.
Boundedness results for multilinear pseudodifferential operators on products of modulation spaces are derived based on ordered integrability conditions on the short-time Fourier transform of the operators' symbols. The flexibility and strength of the introduced methods is demonstrated by their application to the bilinear and trilinear Hilbert transform.
In this paper, we define for the first time the local Morrey-type space associated with ball quasi-Banach function spaces and show the related series of properties. In addition, Hardy-Littlewood maximal operator's boundedness is proved. We investigate nonsmooth decomposition of the local Morrey-type space associated with ball quasi-Banach function spaces via the Hardy local Morrey-type spaces associated with ball quasi-Banach function spaces. And we consider Hardy operator's boundedness.
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 …
