Hua Wang
In this paper, we will study the boundedness properties of intrinsic square functions including the Lusin area integral, Littlewood-Paley $g$-function and $g^*_λ$-function on the weighted Morrey spaces $L^{p,κ}(w)$ for $1<p<\infty$ and $0<κ<1$. The corresponding commutators generated by $BMO(\mathbb R^n)$ functions and intrinsic square functions are also discussed.
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In this paper, we will study the boundedness properties of multilinear Calderon--Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.
In this paper, we study the high-dimensional Hausdorff operators, defined via a general linear mapping $A$, and their commutators on the weighted Morrey spaces in the setting of the Heisenberg group. Particularly, under some assumption on the mapping $A$, we establish their sharp boundedness on the power weighted Morrey spaces.
In this paper, the authors prove the boundedness of commutators generated by the weighted Hardy operator on weighted $λ$-central Morrey space with the weight $ω$ satisfying the doubling condition. Moreover, the authors give the characterization for the weighted $λ$-central Campanato space by introducing a new kind of operator which is related to the commutator of weighted Hardy operator.
In this paper, we first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators $I_α$ in these new Morrey type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators $[b,I_α]$ formed by $b$ and $I_α$ are established. Also we study related problems about two-weight, weak type inequalities for $I_α$ and $[b,I_α]$ in the Morrey type spaces and give partial results.
In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with $BMO(\mathbb R^n)$ functions are discussed. We first show the weighted strong type and weak type estimates of vector-valued intrinsic square functions in the Morrey type spaces. Then we obtain weighted strong type estimates of vector-valued analogues of commutators in Morrey type spaces. In the endpoint case, we establish the weighted weak $L\log L$-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weig…
In this paper, we first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space as special cases. Then we discuss the strong type and weak type estimates for a class of Calderón--Zygmund type operators $T_θ$ in these new Morrey type spaces. Furthermore, the strong type estimate and endpoint estimate of commutators $[b,T_θ]$ formed by $b$ and $T_θ$ are established. Also we study related problems about two-weight, weak type inequalities for $T_θ$ and $[b,T_θ]$ in the Morrey type spaces and give partial results.
Let $T$ be a Calderón-Zygmund singular integral operator. In this paper, we will show some weighted boundedness properties of commutator $[b,T]$ on the weighted Morrey spaces $L^{p,κ}(w)$ under appropriate conditions on the weight $w$, where the symbol $b$ belongs to weighted $BMO$ or Lipschitz space or weighted Lipschitz space.
This paper mainly dedicates to prove a plethora of weighted estimates on Morrey spaces for bilinear fractional integral operators and their general commutators with BMO functions of the form $$B_α(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy,\qquad 0<α<n.$$ We also prove some maximal function control theorems for these operators, that is, the weighted Morrey norm is bounded by the weighted Morrey norm of a natural maximal operator when the weight belongs to $A_{\infty}$. As a corollary, some new weighted estimates for the bilinear maximal function associated to the bilinear Hi…
Let $0<α<n$ and $I_α$ be the fractional integral operator. In this paper, we shall use a unified approach to show some boundedness properties of commutators $[b,I_α]$ on the weighted Morrey spaces $L^{p,κ}(w)$ under appropriate conditions on the weight $w$, where the symbol $b$ belongs to weighted $BMO$ or Lipschitz space or weighted Lipschitz space.
In this paper, we will obtain the weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley $g$-function and $g^*_λ$-function on the weighted Morrey spaces $L^{1,κ}(w)$ for $0<κ<1$ and $w\in A_1$.
We prove that the intrinsic square functions including Lusin area integral and Littlewood-Paley $g^{\ast}_λ$-function as defined by Wilson, are bounded in a class of function spaces include weighted Morrey spaces. The corresponding commutators generated by $BMO$ functions are also considered.
Let $M_{Ω,α}$ and $T_{Ω,α}$ be the fractional maximal and integral operators with rough kernels, where $0<α<n$. In this paper, we shall study the continuity properties of $M_{Ω,α}$ and $T_{Ω,α}$ on the weighted Morrey spaces $L^{p,κ}(w)$. The boundedness of their commutators with BMO functions is also obtained.
In this paper, we establish the boundedness of the commutators of multilinear Hausdorff operators on the product of some weighted Morrey-Herz type spaces with variable exponent with their symbols belong to both Lipschitz space and central BMO space. By these, we generalize and strengthen some previous known results.
In this paper, we will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the weighted Morrey spaces $L^{p,κ}(w)$ when $1\leq p<\infty$, $0<κ<1$, and in the generalized Morrey spaces $L^{p,Φ}$ for $1\le p<\infty$, where $Φ$ is a growth function on $(0,\infty)$ satisfying the doubling condition.
Quantitative Bounds and Compactness for the Commutators of Area Integrals Associated with Self-adjoint Operators on Weighted $L^p$ and Morrey Spaces
研究自伴算子关联的面积积分算子与BMO函数生成的交换子在加权L^p和Morrey空间中的强型估计及紧性。
Let $Ω\in L^q(S^{n-1})$ with $1<q\le\infty$ be homogeneous of degree zero and has mean value zero on $S^{n-1}$. In this paper, we will study the boundedness of homogeneous singular integrals and Marcinkiewicz integrals with rough kernel on the weighted Morrey spaces $L^{p,κ}(w)$ for $q'\le p<\infty$(or $q'<p<\infty$) and $0<κ<1$. We will also prove that the commutator operators formed by a $BMO(\mathbb R^n)$ function $b(x)$ and these rough operators are bounded on the weighted Morrey spaces $L^{p,κ}(w)$ for $q'<p<\infty$ and $0<κ<1$.
Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(\mathbb R^n)$ with Gaussian kernel bound, and let $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n$. In this paper, we will obtain some boundedness properties of commutators $\big[b,L^{-α/2}\big]$ on the weighted Morrey spaces $L^{p,κ}(w)$ when the symbol $b$ belongs to $BMO(\mathbb R^n)$ or homogeneous Lipschitz space.
In this paper we study the boundedness and compactness characterizations of the commutator of Calderón-Zygmund operators $T$ on spaces of homogeneous type $(X,d,μ)$ in the sense of Coifman and Weiss. More precisely, We show that the commutator $[b, T]$ is bounded on weighted Morrey space $L_ω^{p,κ}(X)$ ($κ\in(0,1), ω\in A_{p}(X), 1<p<\infty$) if and only if $b$ is in the BMO space. Moreover, the commutator $[b, T]$ is compact on weighted Morrey space $L_ω^{p,κ}(X)$ ($κ\in(0,1), ω\in A_{p}(X), 1<p<\infty$) if and only if $b$ is in the VMO space.
Let $\mathcal T_α~(0\leqα<n)$ be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let $[b,\mathcal T_α]~(0\leqα<n)$ be the commutators generated by $\mathrm{BMO}(\mathbb R^n)$ functions and $\mathcal T_α$. This paper is concerned with two-weight, weak type norm estimates for these sublinear operators and their commutators on the weighted Morrey and amalgam spaces. Some boundedness criterions for such operators are given, under the assumptions that weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As applications of our ma…
For any $0<α<n$, the homogeneous fractional integral operator $T_{Ω,α}$ is defined by \begin{equation*} T_{Ω,α}f(x)=\int_{\mathbb R^n}\frac{Ω(x-y)}{|x-y|^{n-α}}f(y)\,dy. \end{equation*} In this paper, we prove that if $Ω$ satisfies certain Dini smoothness conditions on $\mathbf{S}^{n-1}$, then $T_{Ω,α}$ is bounded from $L^{p,κ}(w^p,w^q)$ (weighted Morrey space) to $\mathrm{BMO}(\mathbb R^n)$.
In this paper, we will study the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley $g$-function and $g^*_λ$-function on the generalized Morrey spaces $L^{p,Φ}$ for $1\le p<\infty$, where $Φ$ is a growth function on $(0,\infty)$ satisfying the doubling condition. The boundedness of the commutators generated by $BMO(\mathbb R^n)$ functions and intrinsic square functions is also obtained.
In this paper, we consider the boundedness properties of multilinear $θ$-type Calderón--Zygmund operators $T_θ$ recently introduced in the literature. First, we prove strong type and weak type estimates for multilinear $θ$-type Calderón--Zygmund operators on products of weighted Morrey spaces with multiple weights. Then we discuss strong type estimates for both multilinear commutators and iterated commutators of $T_θ$ on products of these spaces with multiple weights. Furthermore, the weak end-point estimates for commutators of $T_θ$ and pointwise multiplication with functions in bounded mean …
In this paper, the boundedness properties of commutators generated by $b$ and intrinsic square functions in the endpoint case are discussed, where $b\in BMO(\mathbb R^n)$. We first establish the weighted weak $L\log L$-type estimates for these commutator operators. Furthermore, we will prove endpoint estimates of commutators generated by $BMO(\mathbb R^n)$functions and intrinsic square functions in the weighted Morrey spaces $L^{1,κ}(w)$ for $0<κ<1$ and $w\in A_1$, and in the generalized Morrey spaces $L^{1,Θ}$, where $Θ$ is a growth function on $(0,+\infty)$ satisfying the doubling condition.
Let $0<γ<n$ and $I_γ$ be the fractional integral operator of order $γ$, $I_γf(x)=\int_{\mathbb R^n}|x-y|^{γ-n}f(y)\,dy$, and let $[b,I_γ]$ be the linear commutator generated by a symbol function $b$ and $I_γ$, $[b,I_γ]f(x)=b(x)\cdot I_γf(x)-I_γ(bf)(x)$. This paper is concerned with two-weight, weak type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain $A_p$-type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator $I_γ$ as well as…
Precompactness in matrix weighted Bourgain-Morrey spaces
本文研究了矩阵加权Bourgain-Morrey空间中预紧集的条件，证明了二进制平均算子的有界性，并揭示了预紧集的必要性。
Let $\mathcal L=-Δ+V$ be a Schrödinger operator, where $Δ$ is the Laplacian on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ for $q\geq d$. The Riesz transform associated with the operator $\mathcal L=-Δ+V$ is denoted by $\mathcal R=\nabla{(-Δ+V)}^{-1/2}$ and the dual Riesz transform is denoted by $\mathcal R^{\ast}=(-Δ+V)^{-1/2}\nabla$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class $RH_q$ for $q\geq d$. Then we will establish the boundedness pro…
Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
研究加权Morrey空间上Hardy-Littlewood极大函数与sharp极大函数交换子的有界性，并据此给出加权Lipschitz函数的新刻画。
Adams type Dunkl Stein-Weiss inequality on Dunkl Morrey spaces on the real line
本文研究了Dunkl分数积分算子在Dunkl-Morrey空间上的加权有界性，推导出Adams型Dunkl Stein-Weiss不等式，并扩展了经典Adams型Stein-Weiss不等式到Dunkl情形。
The multilinear Littlewood-Paley square operators and their commutators on weighted Morrey spaces
本文证明了多重线性Littlewood-Paley平方算子及其交换子在加权Morrey空间上的有界性，并给出了相关g-函数和Marcinkiewicz积分交换子的推论。
Let $\mathcal L=-Δ+V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq3$, where $Δ$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_s$ for $s\geq d/2$. For given $0<α<d$, the fractional integrals associated to the Schrödinger operator $\mathcal L$ is defined by $\mathcal I_α={\mathcal L}^{-α/2}$.Suppose that $b$ is a locally integrable function on $\mathbb R^d$, the commutator generated by $b$ and $\mathcal I_α$ is defined by $[b,\mathcal I_α]f(x)=b(x)\cdot \mathcal I_αf(x)-\mathcal I_α(bf)(x)$. In this paper, we first intr…
Embeddings of Weighted Morrey Spaces
本硕士论文回顾经典Morrey空间定义与性质，推广至加权情形，利用Muckenhoupt权推导加权Morrey空间性质，证明Hardy-Littlewood极大算子有界性，并考察示例权下加权Morrey空间的相互嵌入条件。
In this paper certain $n$-dimensional inequalities are shown to be equivalent to the inequalities in the one-dimensional setting. By this means, embeddings between weighted local Morrey-type spaces are characterized for some ranges of parameters.
Characterizations for multilinear fractional maximal and integral operators and their commutators on generalized weighted Morrey spaces and applications
本文研究了多线性分数次极大与积分算子及其交换子在广义加权Morrey空间上的有界性，证明了两种有界性刻画，并应用于Besov-Morrey空间有界性、嵌入定理及次拉普拉斯算子先验估计。
On matrix weighted Bourgain-Morrey Triebel-Lizorkin spaces
本文研究了矩阵加权Bourgain-Morrey Triebel-Lizorkin空间的定义、等价范数及特征描述，并探讨了伪微分算子在该空间上的有界性。
A description of all the admissible weights similar to the Muckenhoupt class $A_p$ is an open problem for the weighted Morrey spaces. In this paper necessary condition and sufficient condition for two-weight norm inequalities on Morrey spaces to hold are given for the Hardy-Littlewood maximal operator. Necessary and sufficient condition is also verified for the power weights.
Let $\mathcal{L}$ be the Schrödinger operator with potential $V$, that is, $\mathcal L=-Δ+V$, where it is assumed that $V$ satisfies a reverse Hölder inequality. We consider weighted Morrey-Campanato spaces $BMO_{\mathcal L,w}^α(\mathbb R^d)$ and $BLO_{L,w}^α(\mathbb R^d)$ in the Schrödinger setting. We prove that the variation operator $V_σ(\{T_t\}_{t>0})$, $σ>2$, and the oscillation operator $O(\{T_t\}_{t>0}, \{t_j\}_{j\in \mathbb Z})$, where $t_j<t_{j+1}$, $j\in \mathbb Z$, $\lim_{j\rightarrow +\infty}t_j=+\infty$ and $\lim_{j\rightarrow -\infty} t_j=0$, being $T_t=t^k\partial_t^k e^{-t\mat…
In a previous work, "compact versions" of Rubio de Francia's weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of an linear operator from just one space to the full range of weighted Lebesgue spaces, where this operator is bounded. In this paper, we extend these results to the setting of weighted Morrey spaces. As applications, we easily obtain new results on the weighted compactness of commutators of Calderón--Zygmund singular integrals, rough singular integrals and Bochner--Riesz multipliers.
Let $[b,\mathcal T_α]~(0\leqα<n)$ be the commutators generated by $BMO(\mathbb R^n)$ functions and a class of sublinear operators satisfying certain size conditions. The aim of this paper is to study the endpoint estimates of these commutators in the weighted Morrey spaces and in the generalized Morrey spaces, under the assumptions that $[b,\mathcal T_α]~(0\leqα<n)$ satisfy (weighted or unweighted) endpoint inequalities on $\mathbb R^n$ and on bounded domains. Furthermore, as applications of our main results, we will obtain, in the endpoint case, the boundedness properties of many important op…
Sharp bound for m-linear $n$-dimensional Hardy-Littlewood-Polya operator in Morrey space on Heisenberg group
本文在Heisenberg群上的Morrey空间中，获得了m-线性n维Hardy-Littlewood-Pólya算子及Hilbert算子的尖锐界。
We prove that Calderón-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on a wide family of function spaces.
Preduals of weighted homogeneous Bourgain-Morrey Besov-Triebel-Lizorkin spaces associated with operators
研究齐次型空间上与算子相关的加权齐次Bourgain-Morrey Besov-Triebel-Lizorkin空间的预对偶，引入块空间并证明其与单位分解选择无关。
Characterizations for multi-sublinear operators and their commutators on three kinds of generalized weighted Morrey spaces and applications
本文研究多亚线性算子及其交换子在广义加权Morrey空间上有界的条件，给出充分条件及一类算子的充要条件，并应用于Calderón-Zygmund算子、Littlewood-Paley平方算子、伪微分算子和仿积。
A general class of weighted multilinear Hardy-Cesàro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesàro operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.
On Weighted Local Morrey-type and Complementary Local Morrey-Type Spaces
该研究针对加权局部Morrey型与互补局部Morrey型空间，通过一维归约方法结合Cesàro和Copson空间嵌入刻画的最新结果，给出两类空间间连续嵌入的充要条件，覆盖所有有限参数情形。
Weighted Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces associated with operators
本文在齐次型空间上，针对非负自伴算子引入加权Bourgain-Morrey-Besov型与Triebel-Lizorkin型空间，给出连续刻画、原子分子分解，并证明分数幂与谱乘子的有界性。
Let $p\in(1, \infty)$, $κ\in(0, 1)$ and $w\in A_p(\mathbb C).$ In this article, the authors obtain a boundedness (resp., compactness) characterization of the Buerling-Ahlfors commutator $[\mathcal B, b]$ on the weighted Morrey space $L_w^{p,\,κ}(\mathbb C)$ via $\mathrm{BMO}(\mathbb C)$ [resp., $\mathrm{CMO}(\mathbb C)$], where $\mathcal B$ denotes the Buerling-Ahlfors transform and $b\in \mathrm{BMO}(\mathbb C)$ [resp., $\mathrm{CMO}(\mathbb C)$]. Moreover, an application to the Beltrami equation is also given.
In this paper we consider weighted Morrey spaces ${\mathcal M}_{λ, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{λ, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal F}}\left(\frac{1}{|Q|^λ}\int_Q|f|^pw\right)^{1/p},$$ and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy--Littlewood maximal operator on ${\mathcal M}_{λ, {\mathcal F}}^p(w)$. In the case of the global Morrey spaces (when ${\mathcal F}$ is the family of all cubes in ${\mathbb R}^n$) this question is still open. In the case of t…
In this paper, we are interested in the following bilinear fractional integral operator $B\mathcal{I}_α$ defined by \[ B\mathcal{I}_α({f,g})(x)=\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy, \] with $0< α<n$. We prove the weighted boundedness of $B\mathcal{I}_α$ on the Morrey type spaces. Moreover, an Olsen type inequality for $B\mathcal{I}_α$ is also given.
Weighted bounds for a class of singular integral operators in variable exponent Herz-Morrey spaces
本文研究变核奇异积分算子相关算子的有界性，通过球调和展开与卷积算子估计，证明其在带权变指数广义Herz-Morrey空间的有界性并推广已有结果，还建立了算子乘积与伪乘积的范数刻画。
Homogeneous fractional integral operators on weighted Lebesgue, Morrey and Campanato spaces
该研究在Ω满足Dini型光滑性条件下，证明了齐次分数次积分算子T_{Ω,α}在加权Lebesgue空间、Morrey空间到加权Campanato空间的有界性，完善了相关算子的映射性质理论。
The aim of the present paper is to give necessary and sufficient conditions for the boundedness of a general class of multilinear Hausdorff operators that acts on the product of some weighted function spaces with variable exponent such as the weighted Lebesgue, Herz, central Morrey and Morrey-Herz type spaces with variable exponent. Our results improve and generalize some previous known results.
The present article focuses on the bounds of weighted multilinear $p$-adic Hardy operators on the product of Herz spaces and Morrey-Herz spaces. The corresponding norm on both the cases are also obtain.
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 …
In this paper, we give the definability of bilinear singular and fractional integral operators on Morrey-Banach space, as well as their commutators and we prove the boundedness of such operators on Morrey-Banach spaces. Moreover, the necessary condition for BMO via the bounedness of bilinear commutators on Morrey-Banach space is also given. As a application of our main results, we get the necessary conditions for BMO via the bounedness of bilinear integral operators on weighted Morrey space and Morrey space with variable exponents. Finally, we obtain the boundedness of bilinear C-Z operator on…
We prove a plethora of boundedness property of the Adams type for bilinear fractional integral operators of the form $$B_α(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy,\qquad 0<α<n.$$ For $1<t\leq s<\infty$, we prove the non-weighted case through the known Adams type result. And we show that these results of Adams type is optimal. For $0<t\leq s<\infty$ and $0<t\leq1$, we obtain new result of a weighted theory describing Morrey boundedness of above form operators if two weights $(v,\vec{w})$ satisfy $$ [v,\vec{w}]_{t,\vec{q}/{a}}^{r,as}=\mathop{\sup_{Q,Q^{\prime}\in\mathscr{D}…
Commutators for certain fractional type operators on weighted spaces and Orlicz-Morrey spaces
本文研究一类推广Riesz位势的分数型积分算子，得到其与BMO型函数交换子的加权Lp、加权Hardy估计，及其与Campanato空间函数交换子在Orlicz-Morrey空间的有界性与m=1、A=I时的紧性。
Sobolev--Morrey Spaces and Divergence-Form Degenerate Second-Order Elliptic Equations on Domains with Higher Co-Dimensional Boundaries
研究\(\mathbb{R}^n\)中高余维边界区域上加权齐次Sobolev - Morrey空间，建立实变理论，应用迹和延拓算子有界性，获散度型退化二阶椭圆方程Dirichlet问题加权先验估计。
Boundedness of the multilinear integral operators on Heisenberg group
本文研究海森堡群上多线性积分算子的有界性，获得Lebesgue空间和双幂权Morrey空间的锐利常数，并证明$A_p$加权Morrey空间上的有界性。
Let $Ω$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mathcal{M}_Ω$ be the higher-dimensional Marcinkiewicz integral associated with $Ω$. In this paper, the author considers the complete continuity on weighted $L^p(\mathbb{R}^n)$ spaces with $A_p(\mathbb{R}^n)$ weights, weighted Morrey spaces with $A_p(\mathbb{R}^n)$ weights, for the commutator generated by ${\rm CMO}(\mathbb{R}^n)$ functions and $\mathcal{M}_Ω$ when $Ω$ satisfies certain size conditions.
We obtain sufficient and necessary conditions on weight functions $s_1(t),\ldots,s_m(t)$ and $ψ(t)$ so that the weighted multilinear Hardy-Cesàro operator \[(f_1,\ldots,f_m)\mapsto \int_{[0,1]^n}\left(\prod_{k=1}^nf_k\left(s_k(t) x\right)\right)ψ(t)dt \] is bounded from $\dot{K}^{α_1, p_1}_{q_1}(ω_1)\times \cdots \times\dot{K}^{α_m, p_m}_{q_m}(ω_m)$ to $\dot{K}^{α, p}_{q}(ω)$ and from $M\dot{K}^{α_1, λ_1}_{p_1,q_1}(ω_1)\times \cdots \times M\dot{K}^{α_m, λ_m}_{p_m,q_m}(ω_m)$ to $M\dot{K}^{α, λ}_{p,q}(ω)$. The sharp bounds are also obtained and these results hold for both cases $0<p<1$ and $1\l…
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