In this paper, the main aim is to consider the boundedness of commutators of multilinear Calderón-Zygmund operators with Lipschitz functions in the context of the variable exponent Lebesgue spaces. Furthermore, the variable versions of the Lipschitz spaces are also discussed, and with the help of a key tool of a pointwise estimate involving the sharp maximal operator of the multilinear fractional commutator and certain associated maximal operators.
We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators including Calderón-Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order multilinear setting. The proof involves a pointwise sparse domination of multilinear commutators.
We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BMO functions via the boundedness of certain paraproducts and also via the boundedness of the commutators of multilinear Haar Multipliers and paraproduct operators.
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.
We extend the recently much-studied two-weight commutator estimates to the multilinear setting. In contrast to previous results, our result respects the multilinear nature of the problem fully and is formulated with the genuinely multilinear weights.
In this article, we show that multilinear fractional type operators are bounded from product Hardy spaces with variable exponents into Lebesgue spaces with variable exponents via the atomic decomposition theory. We also study continuity properties of commutators of multilinear fractional type operators on product of certain Hardy spaces with variable exponents.
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 …
This note is devoted to establishing two-weight estimates for commutators of singular integrals. We combine multilinearity with product spaces. A new type of two-weight extrapolation result is used to yield the quasi-Banach range of estimates.
In this paper, the behavior for commutators of a class of bilinear singular integral operator associated with non-smooth kernels on the products of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO functions, compactness of the commutators is proved.
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.
In this paper, we study properties of the bilinear multiplier space. We give a necessary condition for a continuous integrable function to be a bilinear multiplier on variable exponent Lebesgue spaces. And we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.
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 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 ${\cal T}_ϕ$.
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.
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, 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.
Iterated commutators of multilinear Calderon-Zygmund operators and pointwise multiplication with functions in $BMO$ are studied in products of Lebesgue spaces. Both strong type and weak end-point estimates are obtained, including weighted results involving the vectors weights of the multilinear Calderon-Zygmund theory recently introduced in the literature. Some better than expected estimates for certain multilinear operators are presented too.
In this paper, we investigate the multilinear boundedness properties of the higher ($n$-th) order Calderón commutator for dimensions larger than two. We establish all multilinear endpoint estimates for the target space $L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, including that Calderón commutator maps the product of Lorentz spaces $L^{d,1}(\mathbb{R}^d)\times\cdots\times L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, which is the higher dimensional nontrivial generalization of the endpoint estimate that the $n$-th order Calderón commutator maps $L^{1}…
Let $T$ be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of $\mathbb R^n$. We assume that the associated kernel of $T$ satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for $T$ and the boundedness of the commutator of $T$ with BMO functions on products of weighted Lebesgue spaces of $\mathbb R^n$. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of the…
We deal with the boundedness of the multilinear fractional integral operator $I_{γ,m}$ from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We characterize the classes of weights for which the problem described above holds and show the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region. We further exhibit examples…
Smoothness properties related to several commutators of fractional operators for values of $\boldsymbol{p}$ beyond the extreme in the multilinear setting
Let $T_{\vec{b}}$ and $T_{Πb}$ be the commutators in the $j$-th entry and iterated commutators of the multilinear Calderón-Zygmund operators, respectively. It was well-known that $T_{\vec{b}}$ and $T_{Πb}$ were not of weak type $(1,1)$ and $(H^1, L^1)$, but they did satisfy certain endpoint $L\log L$ type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that $T_{\vec{b}}$ and $T_{Πb}$ are bounded from product Hardy space $H^1\times\cdot\cdot\cdot\times H^1$ to weak $L^{\frac{1}{m},\infty}$ space, whenever the kernel satisfies a class of Dini type condit…
We deal with the boundedness properties of higher order commutators related to some generalizations of the multilinear fractional integral operator of order $m$, $I_α^m$, from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces, extending some previous estimates for the linear case. Our study includes two different types of commutators and sufficient conditions on the weights in order to guarantee the continuity properties described above. We also exhibit the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defini…
We prove mixed inequalities for commutators of Calderón-Zygmund operators (CZO) with multilinear symbols. Concretely, let $m\in\mathbb{N}$ and $\mathbf{b}=(b_1,b_2,\dots, b_m)$ be a vectorial symbol such that each component $b_i\in \mathrm{Osc}_{\mathrm{exp}\, L^{r_i}}$, with $r_i\geq 1$. If $u\in A_1$ and $v\in A_\infty(u)$ we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\] holds for every $t>0$, where $Φ(t)=t(1+\log^+t)^r$, with $1/r=\sum_{i=1}^…
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
Given an $m$-tuple of weights $\vec{v}=(v_1,\dots,v_m)$, we characterize the classes of pairs $(w,\vec{v})$ involved with the boundedness properties of the multilinear fractional integral operator from $\prod_{i=1}^mL^{p_i}\left(v_i^{p_i}\right)$ into suitable Lipschitz spaces associated to a parameter $δ$, $\mathcal{L}_w(δ)$. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We emphasize the study related to the range of the parameters involved with the problem described above, which is optimal in the se…
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 [C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6(1):1-15, 1999], the following type of multilinear fractional integral \[ \int_{\mathbb{R}^{mn}} \frac{f_1(l_1(x_1,\ldots,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots,x_m,x))}{(|x_1|+\ldots+|x_m|)^λ} dx_1\ldots dx_m \] was studied, where $l_i$ are linear maps from $\mathbb{R}^{(m+1)n}$ to $\mathbb{R}^n$ satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from $L^{p_1}\times \ldots \times L^{p_{m+1}}$ to $L^q$ when the indices satisfy the homogeneity c…
Let $0<α<n$ and $M_α$ be the fractional maximal function. The nonlinear commutator of $M_α$ and a locally integrable function $b$ is given by $[b,M_α](f)=bM_α(f)-M_α(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_α]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\rn)$ spaces are obtained.
We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and Lipschitz spaces via the weighted boundedness of commutators of multilinear Calderón-Zygmund operators with the genuinely multilinear weights.
We prove that a generalized Fefferman-Phong type condition on a pair of weights $u$ and $v$ is sufficient for the boundedness of the commutators of potential type operators from $L^{p(\cdot)}_v$ into $L^{q(\cdot)}_u$. We also give an improvement of this result in the sense that we not only consider a variable version of power bump conditions, but also weaker norms related to Musielak-Orlicz functions. We consider a wider class of symbols including Lipschitz symbols and some generalizations.
In this paper, we investigate the weighted vector-valued bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\,\mathbb{R}^n,w_1)\times\dots\times L^{p_m}(l^{q_m};\,\mathbb{R}^n,w_m)$ to $L^{p}(l^q;\,\mathbb{R}^n,ν_{\vec{w}})$, with $p_1,\dots,p_m\in (1,\,\infty)$ and $1/p=1/p_1+\dots+1/p_m$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators.
We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in $L^r$-based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We consider the case $1<r\leq 2$ and we characterize boundedness in terms of inequalities relating the Lebesgue indices (or Hardy indices), the dimension, and the regularity and integrability indices of the Sobolev space. The case $r>2$ cannot be handled by known techniques and remains open. Our result not only extends but also establishes the sharpness of previous r…
This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms.
Let $m\in \mathbb{N}$ and $\vec{b}=(b_{1},\cdots,b_{m})$ be a collection of locally integrable functions. It is proved that $b_{1},b_{2},\cdots, b_{m}\in BMO$ if and only if $$\sup_{Q}\frac{1}{|Q|^{m}}\int_{Q^{m}}\Big|\sum_{i=1}^{m}\big(b_{i}(x_{i})-(b_{i})_{Q}\big)\Big|d\vec{x}<\infty,$$ where $(b_{i})_{Q}=\frac{1}{|Q|}\int_{Q}b_{i}(x)dx$. As an application, we show that if the linear commutator of certain multilinear Calderón-Zygmund operator $[Σ\vec{b},T]$ is bounded from $L^{p_{1}}\times\cdots\times L^{p_{m}}$ to $L^{p}$ with $\sum_{i=1}^{m}1/p_{i}=1/p$ and $1<p,p_{1},\cdots,p_{m}<\infty$,…
Let $(X,d,μ)$ be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling measure conditions. In this paper, the boundedness of multilinear fractional integral operator in this setting is proved. Via a sharp maximal operator, the boundedness of commutators generated by multilinear fractional integral operator with $RBMO(μ)$ function on non-homogeneous metric measure spaces in Lebesgue spaces is obtained.
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…
Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered. Lebesgue space norm inequalities are established for multilinear integral operators of Calderon-Zygmund type which incorporate oscillatory factors exp(iP), where P is a real-valued polynomial with large coefficients. A related problem concerning upper bounds for measures of sublevel sets is solved.
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.
We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander $S^{m}_{ρ, δ}$ classes. These results are new in the case $ρ< 1$, that is, outwith the scope of multilinear Calderón-Zygmund theory.
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.
We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…
We give a weak factorization proof of the Hardy space $H^{p}(\mathbb{R}^{n})$ in the multilinear setting, for $\frac{n}{n+1} < p <1$. As a consequence, we obtain a characterization of the boundedness of the commutator $[b,T]$ from $L^{r_{1}}(\mathbb R^n) \text{~x ... x~} L^{r_{m}} (\mathbb R^n) \text{ to } L^{q^\prime} (\mathbb R^n)$, where $b \in \text{Lip}_α(\mathbb R^n)$, and $\fracα{n} = \sum_{i=1}^{m} \frac{1}{r_{i}} +\frac{1}{q} - 1$.
