In this work we study boundedness of Littlewood-Paley-Stein square func- tions associated to multilinear operators. We prove weighted Lebesgue space bounds for square functions under relaxed regularity and cancellation conditions that are independent of weights, which is a new result even in the linear case. For a class of multilinear convolu- tion operators, we prove necessary and sufficient conditions for weighted Lebesgue space bounds. Using extrapolation theory, we extend weighted bounds in the multilinear setting for Lebesgue spaces with index smaller than one.
In this paper, weighted norm inequalities with $A_p$ weights are established for the multilinear singular integral operators whose kernels satisfy $L^{r'}$-Hörmander regularity condition. As applications, we recover a weighted estimate for the multilinear Fourier multiplier obtained by Fujita and Tomita, and obtain several new weighted estimates for the multilinear Fourier multiplier as well.
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 prove that if a multilinear operator $\mathcal{T}$ and its multilinear commutator $\mathcal{T}_{Σ\vec{b}}$ and iterated commutator $\mathcal{T}_{Π\vec{b}}$ for $\vec{b}\in(\mathbb{R}^n)^m$ are bounded on product weighted Lebesgue space, then $\mathcal{T}$, $\mathcal{T}_{Σ\vec{b}}$ and $\mathcal{T}_{Π\vec{b}}$ are also bounded on product weighted Amalgam space. As its applications, we show that multilinear Littlewood-Paley functions and multilinear Marcinkiewicz integral functions with kernels of convolution type and non-convolution type, and their multilinear commutators and …
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.
In this paper, we first introduce the new class of multiple weights $A^\vc_{\vec{p}}$ which is larger than the class of multiple weights in \cite{LOPTG}. Then, using this class of weights, we study the weighted norm inequalities for certain classes of multilinear operators and their commutators with new BMO functions introduced by \cite{BHS1}. Finally, we show that some multilinear pseudodifferential operators fall within the scope of the theory obtained in this paper.
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 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 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, the necessity theory for commutators of multilinear singular integral operators on weighted Lebesgue spaces is investigated. The results relax the restriction of the weights class to the general multiple weights, which can be regarded as an essential improvement of \cite{ChafCruz2018,GLW2020}. Our approach elaborates on a commonly expanding the kernel locally by Fourier series, recovering many known results but yielding also numerous new ones. In particular, we answer the question about the necessity theory of the iterated commutators of the multilinear singular integral operato…
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 article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely $\textbf{SG}$-class.
Let $μ_{Ω,\vec{b}}$ be the multilinear commutator generalized by $μ_Ω$, the $n$-dimensional Marcinkiewicz integral, with $\Osc_{\exp L^{^τ}}(\R^{n})$ functions for $τ\ge 1$, where $\Osc_{\exp L^{^τ}}(\R^{n})$ is a space of Orlicz type satisfying that $\Osc_{\exp L^{^τ}}(\R^{n})=\BMO(\R^{n})$ if $τ=1$ and $\Osc_{\exp L^{^τ}}(\R^{n})\subset\BMO(\R^{n})$ if $τ>1$. The authors establish the weighted weak $L\log L$-type estimates for $μ_{Ω,\vec{b}}$ when $Ω$ satisfies a kind of Dini conditions.
Let $μ_{Ω,\vec{b}}$ be the multilinear commutator generalized by $μ_Ω$, the $n$-dimensional Marcinkiewicz integral with the bounded kernel, and $b_{j}\in \Osc_{\exp L^{r_{j}}}(1\le j\le m)$. In this paper, the following weighted inequalities are proved for $ω\in A_{\infty}$ and $0<p<\infty$, $$\|μ_Ω(f)\|_{L^{p}(ω)}\leq C\|M(f)\|_{L^{p}(ω)}, \ \ \|μ_{Ω,\vec{b}}(f)\|_{L^{p}(ω)}\leq C\|M_{L(\log L)^{1/r}}(f)\|_{L^{p}(ω)}.$$ The weighted weak $L(\log L)^{1/r}$ -type estimate is also established when $p=1$ and $ω\in A_{1}$.
We obtain restriction results of K. de Leeuw's type for maximal operators defined through multilinear Fourier multipliers of either strong or weak type acting on weighted Lebesgue spaces. We give some application of our development. In particular we obtain periodic weighted results for Coifman-Meyer, Hörmander and Hörmander-Mihlin type multilinear multipliers.
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.
Characterizations for multilinear fractional maximal and integral operators and their commutators on generalized weighted Morrey spaces and applications
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.
The aim of this paper is to get the product Lp-estimates, weighted estimates and two-weighted estimates for rough multilinear fractional integral operators and rough multi-sublinear fractional maximal operators, respectively. The author also studies two-weighted weak type estimate on product Lp (Rn) for rough multi-sublinear fractional maximal operators. In fact, this article is the rough kernel versions of [4, 5]' s results.
We give a constructive proof of the factorization theorem for the classical Hardy space in terms of fractional integral operator. Moreover, the result is extended to the multilinear case and weighted case. As an application, we obtain the characterization of $BMO$ via the weighted boundedness of commutators of the multilinear fractional integral operator, without individual conditions on the weights class.
Let $S_α$ be the multilinear square function defined on the cone with aperture $α\geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_α$. We first obtain a sharp weighted estimate in terms of aperture $α$ and $\vec{w} \in A_{\vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer's conjecture, for which a Coifman-Fefferman inequality with the precise…
We initiate the study of a duality theory which applies to norm inequalities for pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey-Nikisin-Stein theory of factorisation of operators, and present a fully multilinear version of Maurey's fundamental theorem on factorisation of operators through $L^1$. The development of the theory involves convex optimisation and minimax theor…
A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Certain bilinear interpolation results between distributional estimates, such as that obtained for the multivariable strong maximal fu…
We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on $\ell^r$-valued extensions of linear operators. We show that for certain $1 \leq p, q_1, \dots, q_m, r \leq \infty$, there is a constant $C\geq 0$ such that for every bounded multilinear operator $T\colon L^{q_1}(μ_1) \times \cdots \times L^{q_m}(μ_m) \to L^p(ν)$ and functions $\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(μ_1), \dots, \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(μ_m)$, the following inequality holds \begin{equation}\label{MZ ineq abstract} (1) \quad \quad \left\Vert \left(\sum_{k_1, \dots…
We define new generalized Herz spaces having weight and variable exponent, that is, weighted Herz spaces with variable exponent. We prove the boundedness of an intrinsic square function on those spaces under proper assumptions on each exponent and weight.
A generalization with singular weights of Moore-Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.
It is proved that, up to isomorphism, there are only two directed trees that admit a hyponormal weighted shift with nonzero weights whose square has trivial domain. These are precisely those enumerable directed trees, one with root, the other without, whose every vertex has enumerable set of successors.
It is shown that for a bounded weighted bilateral shift $T$ acting on $\ell_p(\Z)$ for $1\leq p\leq 2$ supercyclicity of $T$, weak supercyclicity of $T$, cyclicity of $T\oplus T$ and cyclicity of $T^2$ are equivalent. A new sufficient condition for cyclicity of a weighted bilateral shift is proved, which implies, in particular, that any compact weighted bilateral shift is cyclic.
The paper contains the proof of $L^p$-weighted norm inequalities for both, martingales square functions and the classical square functions in harmonic analysis of Littlewood-Paley and Lusin. Furthermore, the bounds are completely explicit and are optimal not only on the dependence of the characteristics of the weight but also on the dependance on $p$, as $p\to\infty$. The proof rests on Bellman function method: the estimates are deduced from the existence of an appropriate and rather complicated function of four variables.
