In this paper we approach the two weighted boundedness of commutators via matrix weights. This approach provides both a sufficient and a necessary condition for the two weighted boundedness of commutators with an arbitrary linear operator in terms of one matrix weighted norm inequalities for this operator. Furthermore, using this approach, we surprisingly provide conditions that almost characterize the two matrix weighted boundedness of commutators with CZOs and completely arbitrary matrix weights, which is even new in the fully scalar one weighted setting. Finally, our method allows us to extend the two weighted Holmes/Lacey/Wick results to the fully matrix setting (two matrix weights and a matrix symbol), completing a line of research initiated by the first two authors.
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 article, we investigate the boundedness properties of the multilinear dyadic paraproduct operators in the weighted setting. We also obtain weighted estimates for the multilinear Haar multipliers and their commutators with dyadic BMO functions.
In this article, we study the commutators of Hausdor? operator and establish their boundedness on weighted Herz space in the setting of Heisenberg group.
Commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact on appropriate products of weighted Lebesgue spaces.
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…
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.
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 we prove two matrix weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a matrix symbol. More precisely, we extend the recent results of the second author, Pott, and Treil on two matrix weighted norm inequalities for commutators of Calderon-Zygmund operators and multiplication by a matrix symbol to the fractional integral operator setting. In particular, we completely extend the fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted setting with a matrix
In this paper we prove several weighted estimates for bilinear fractional integral operators and their commutators with BMO functions. We also prove maximal function control theorem for these operators, that is, we prove the weighted $L^p$ norm is bounded by the weighted $L^p$ norm of a natural maximal operator when the weight belongs to $A_\infty$. As a corollary we are able to obtain new weighted estimates for the bilinear maximal function associated to the bilinear Hilbert transform.
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.
This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-Hörmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such…
The recent proof of the sharp weighted bound for Calderón-Zygmund operators has led to much investigation in sharp mixed bounds for operators and commutators, that is, a sharp weighted bound that is a product of at least two different $A_p$ weight constants. The reason why these are sought after is that the product will be strictly smaller than the original one-constant bound. We prove a variety of these bounds in spaces of homogeneous type, using the new techniques of Lerner, for both operators and commutators.
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, 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.
We prove weighted strong inequalities for the multilinear potential operator ${\cal T}_ϕ$ and its commutator, where the kernel $ϕ$ satisfies certain growth condition. For these operators we also obtain Fefferman-Stein type inequalities and Coifman type estimates. On the other hand we prove weighted weak type inequalities for the multilinear maximal operator $\mathcal{M}_{φ,B}$ associated to a essentially nondecreasing function $φ$ and to a submultiplicative Young function $B$. This result allows us to obtain a weighted weak type inequality for the operator ${\cal T}_ϕ$.
This paper is dedicated to study weighted $L^p$ inequalities for pseudo-differential operators with amplitudes and their commutators by using the new class of weights $A_p^\vc$ and the new BMO function space BMO$_\vc$ which are larger than the Muckenhoupt class of weights $A_p$ and classical BMO space BMO, respectively. The obtained results therefore improve substantially some well-known results.
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 $Ω$ 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.
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.
Let $\mathcal{L}=-Δ+\mathit{V}(x)$ be a Schrödinger operator, where $Δ$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse Hölder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schrödinger operators, which include Riesz transforms, standard Calderón-Zygmund operatos and Littlewood-Paley functions. These results generalize substantially some well-know results.
Let $L=-Δ+V$ with non-negative potential $V$ satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to $L$ such as Riesz transforms and fractional integrals with the new BMO functions introduced in \cite{BHS1} on the weighted spaces $L^p(w)$ where $w$ belongs to the new classes of weights introduced by \cite{BHS2}.
In this paper, we first introduce several new classes of weighted amalgam spaces. Then we discuss both strong type and weak type estimates for certain multilinear $θ$-type Calderón--Zygmund operators $T_θ$ recently introduced in the literature on products of these spaces with multiple weights. Furthermore, the strong type and weak end-point estimates for both multilinear commutators and iterated commutators of $T_θ$ and pointwise multiplication with BMO functions are established as well.
In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we discuss the strong type and weak type estimates for a class of Calderón--Zygmund type operators $T_θ$ in these new weighted spaces. Furthermore, the strong type estimate and endpoint estimate of linear 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 weighted amalgam spaces and give some results.
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.
Let H be the Hermite operator -Δ+|x|^2 on \mathbb{R}^n. We prove a weighted L^2 estimate of the maximal commutator operator \sup_{R>0}|[b, S_R^λ(H)](f)|, where [b, S_R^λ(H)](f) = bS_R^λ(H) f - S_R^λ(H)(bf) is the commutator of a BMO function b and the Bochner-Riesz means S_R^λ(H) for the Hermite operator H. As an application, we obtain the almost everywhere convergence of [b, S_R^λ(H)](f) for large λand f\in L^p(\mathbb{R}^n).
In this paper we study the commutators of fractional type integral operators. This operators are given by kernels of theform $$K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),$$ where $A_i$ are invertibles matrices and each $k_i$ satisfies a fractional size condition and generalized fractional Hörmander condition. We obtain weighted Coifman estimates, weighted $L^p(w^p)$ - $L^q(w^q)$ estimates and weighted BMO estimates. We also give a two weight strong estimate for pair of weights of the form $(u,Su)$ where $u$ is an arbitrary non-negative function and $S$ is a maximal operator depending on th…
In this paper we characterize the two matrix weighted boundedness of commutators with any of the Riesz transforms (when both are matrix A${}_p$ weights) in terms of a natural two matrix weighted BMO space. Furthermore, we identify this BMO space when $p = 2$ as the dual of a natural two matrix weighted H${}^1$ space, and use our commutator result to provide a converse to Bloom's matrix A${}_2$ theorem, which as a very special case proves Buckley's summation condition for matrix A${}_2$ weights. Finally, we use our results to prove a matrix weighted John-Nirenberg inequality, and we also briefl…
In this paper, we gave a weighted compactness theory for the generalized commutators of vecotor-valued multilinear Calderón-Zygmund operators. This was done by establishing a weighted Fréchet-Kolmogorov theorem, which holds for weights not merely in $A_\infty$. This weighted theory also extends the previous known unsatisfactory results in the terms of relaxing the index to the natural range. As consequences, we not only obtained the weighted compactness theory for the commutators of multilinear Calderón-Zygmund operators, but also extended the same results to the commutators of multilinear Lit…
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.
We present a unified method to obtain weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones.
In this paper we prove two-weighted norm estimates for higher order commutator of singular integral and fractional type operators between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-trivial weights in the optimal region satisfying the conditions required.
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…
In this paper, we introduce a new weighted Hardy-Cesàro operator defined by $U_{ψ,s}f(x)=\int\limits_0^1 f(s(t)\cdot x) ψ(t)dt$, which is associated to the parameter curve $s(t,x)=s(t)x$. Under certain conditions on $s(t)$ and on an absolutely homogeneous weight function $ω$, we characterize the weight function $ψ$ such that $U_{ψ,s}$ is bounded on $L^p(ω)$, $BMO(ω)$. The corresponding operator norms are worked out too. These results extend the ones of Jie Xiao \cite{xiao}. We also give a sufficient and a necessary condition on the weight function $ψ$, which ensure the boundedness of the commu…
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.
In recent years, it has been well understood that a Calderón-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a locally integrable function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.
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…
In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we deal with the vector-valued intrinsic square functions, which are given by \begin{equation*} \mathcal S_γ(\vec{f})(x) = \Bigg(\sum_{j=1}^\infty \big|\mathcal S_γ(f_j)(x) \big|^2\Bigg)^{1/2}, \end{equation*} where $0<γ\leq1$ and \begin{equation*} \mathcal S_γ(f_j)(x) = \left(\iint_{Γ(x)} \Big[\sup_{φ\in{\mathcal C}_γ} \big|φ_t*f_j(y) \big|\Big]^2 \frac{dydt}{t^{n+1}}\right)^{1/2}, \quad j=1,2,\dots. \end{equation*} In his fundamental work, Wilson established strong-type and weak-type estimates for vector-valued…
In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new $CMO(\mathbb{R}^n)$ characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.
Let $K$ be the Calderón-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$. Define the commutator associated with $K$ and $a\in L^\infty(\mathbb{R}^d)$ by \[ T_af(x)=p.v. \int K(x-y)m_{x,y}a\cdot f(y)dy. \] Recently, Grafakos and Honz\'ık [5] proved that $T_a$ is of weak type (1,1) for $d=2$. In this paper, we show that $T_a$ is also weighted weak type (1,1) with the weight $|x|^α\,(-2<α<0)$ for $d=2$. Moreover, we prove that $T_a$ is bounded on weighted $L^p(\mathbb{R}^d)\,(1<p<\infty)$ for all $d\ge2$.
We consider weighted $L^p(w)$ boundedness ($1<p<\infty $ and $w$ a Muckenhoupt $A_p$ weight) of the Calderón commutator $\mathcal C_Ω$ associated with rough homogeneous kernel, under the condition $Ω\in L^q(\mathbb S^{n-1})$ for $q_0<q\leq\infty$ with $q_0$ a fixed constant depending on $w$. Comparing to the previous related known results (assuming $Ω\in L^\infty(\mathbb S^{n-1})$), our result for $Ω\in L^q(\mathbb S^{n-1})$ with $q$ in the range $(q_0,\infty)$ is new. We also obtain a quantitative weighted bound for this $\mathcal C_Ω$ on $L^p(w)$, which is the best known quantitative result …
We establish weighted inequalities for $BMO$ commutators of sublinear operators for all $0<p<\infty$. For weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse Hölder condition, we prove that $\bullet$ commutators $T_b$, which are bounded on $L^p$ with $1<p<\infty$, are bounded from some subspaces of $L^p_w$ to $L^p_w$ and to themselves for all $0<p<\infty$, these are applied to the commutators of singular integral operators and Hardy-Littelwood maximal operator, et.al, which are known to fail to be bounded from $H^1$ to $L^1$ and whose estimate has been open …
Let $T$ be a multilinear Calderón-Zygmund operator of type $ω$ with $ω(t)$ being nondecreasing and satisfying a kind of Dini's type condition. Let $T_{Π\vec{b}}$ be the iterated commutators of $T$ with $BMO$ functions. The weighted strong and weak $L(\log{L})$-type endpoint estimates for $T_{Π\vec{b}}$ with multiple weights are established. Some boundedness properties on weighted variable exponent Lebesgue spaces are also obtained. As applications, multiple weighted estimates for iterated commutators of paraproducts and bilinear pseudo-differential operators with mild regularity are given.
Let $B$ be a locally integrable matrix function, $W$ a matrix A${}_p$ weight with $1 < p < \infty$, and $T$ be any of the Riesz transforms. We will characterize the boundedness of the commutator $[T, B]$ on $L^p(W)$ in terms of the membership of $B$ in a natural matrix weighted BMO space. To do this, we will characterize the boundedness of dyadic paraproducts on $L^p(W)$ via a new matrix weighted Carleson embedding theorem. Finally, we will use some of the ideas from these proofs to (among other things) obtain quantitative weighted norm inequalities for these operators and also use them to pro…
In this paper, we first introduce some new classes of weighted amalgam spaces. Then we give the weighted strong-type and weak-type estimates for fractional integral operators $I_γ$ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators $[b,I_γ]$ generated by $b$ and $I_γ$ are established as well. In addition, we are going to study related problems about two-weight, weak type inequalities for $I_γ$ and $[b,I_γ]$ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove nor…
In this paper we obtain a pointwise sparse domination for generalized Hörmander operators and also for iterated commutators with those operators. As a particular case of our result we obtain a extension of the sparse domination for commutators obtained by A. Lerner, S. Ombrosi and the second author to iterated commutators. Relying upon that sparse domination a number of quantitative estimates such as Coifman-Fefferman estimates, strong type estimates, and endpoint estimates that improve and complete results in papers by M. Lorente, J.M. Martell, C. Pérez, M.S. Riveros and A. de la Torre are ob…
In this paper, the following iterated commutators $T_{*,Πb}$ of maximal operator for multilinear singular integral operators and $I_{α, Πb}$ of multilinear fractional integral operator are introduced and studied $$\aligned T_{*,Πb}(\vec{f})(x)&=\sup_{δ>0}\bigg|[b_1,[b_2,...[b_{m-1},[b_m,T_δ]_m]_{m-1}...]_2]_1 (\vec{f})(x)\bigg|,$$ $$\aligned I_{α, Πb}(\vec{f})(x)&=[b_1,[b_2,...[b_{m-1},[b_m,I_α]_m]_{m-1}...]_2]_1 (\vec{f})(x),$$ where $T_δ$ are the smooth truncations of the multilinear singular integral operators and $I_α$ is the multilinear fractional integral operator, $b_i\in BMO$ for $i=1,…
In this paper, the weighted Lp boundedness of multilinear commutators and iterated commutators of multilinear singular integral operators with generalized kernels is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels.
In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calderón commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the $L^p(\mathbb{R}^d,w)$ space, including some peculiar endpoint estimates of the higher dimensional Calderón commutator.
In this paper, the authors establish some weighted estimates for the Calderón commutator defined by \begin{eqnarray*} &&\mathcal{C}_{m+1,\,A}(a_1,\dots,a_{m};f)(x) &&\quad={\rm p.\,v.}\,\int_{\mathbb{R}}\frac{P_2(A;\,x,\,y)\prod_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y, \end{eqnarray*} with $P_2(A;\,x,\,y)=A(x)-A(y)-A'(y)(x-y)$. Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from $L^{p_1}(\mathbb{R},w_1)$ $\times\dots\times L^{p_m}(\mathbb{R},w_m)$ to $L^{p}(\mathbb{R},ν_{\vec{w}})$, with $p_1,\dots,p_m \in (1,\,\infty)$, $1/p=1…
We characterize the boundedness of the commutators $[b, T]$ with biparameter Journé operators $T$ in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little $bmo$ norm of the symbol $b$. Specifically, if $μ$ and $λ$ are biparameter $A_p$ weights, $ν:= μ^{1/p}λ^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(ν)$, then we prove a lower bound and testing condition $\|b\|_{bmo(ν)} \lesssim \sup \| [b, R_k^1 R_l^2]: L^p(μ) \rightarrow L^p(λ)\|$, where $R_k^1$ and $R_l^2$ are Riesz transforms acting in each variable. Further, we prove that for suc…
In this paper we obtain some mixed means and weighted $L^p$ estimates for the commutators generating $r$ order central integral means operators with $CMO$ functions.
