Multilinear square functions and multiple weights

2019 ◽  
Vol 124 (1) ◽  
pp. 149-160
Author(s):  
Loukas Grafakos ◽  
Parasar Mohanty ◽  
Saurabh Shrivastava
Keyword(s):  
Affirmative Answer ◽  
Square Functions ◽  
Weighted Estimates ◽  
Multiple Weights

In this paper we prove weighted estimates for a class of smooth multilinear square functions with respect to multilinear $A_{\vec P}$ weights. In particular, we establish weighted estimates for the smooth multilinear square functions associated with disjoint cubes of equivalent side-lengths. As a consequence, for this particular class of multilinear square functions, we provide an affirmative answer to a question raised by Benea and Bernicot (Forum Math. Sigma 4, 2016, e26) about unweighted estimates for smooth bilinear square functions.

scholarly journals Weighted Vector-Valued Inequalities for a Class of Multilinear Singular Integral Operators

2018 ◽  
Vol 61 (2) ◽  
pp. 413-436 ◽  
Author(s):  
Guoen Hu ◽  
Kangwei Li
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Maximal Operators ◽  
Weighted Estimates ◽  
Multilinear Singular Integral ◽  
Multiple Weights ◽  
Vector Valued

AbstractIn this paper, some weighted vector-valued inequalities with multiple weights $A_{\vec P}$ (ℝmn)are established for a class of multilinear singular integral operators. The weighted estimates for the multi(sub)linear maximal operators which control the multilinear singular integral operators are also considered.

scholarly journals Sharp Weighted Bounds for Multilinear Fractional Type Operators Associated with Bergman Projection

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Juan Zhang ◽  
Senhua Lan ◽  
Qingying Xue
Keyword(s):  
Maximal Function ◽  
Bergman Projection ◽  
Weighted Estimates ◽  
Fractional Maximal Function ◽  
Multiple Weights ◽  
Fractional Type

We first introduce the multiple weights which are suitable for the study of Bergman type operators. Then, we give the sharp weighted estimates for multilinear fractional Bergman operators and fractional maximal function.

scholarly journals Conical square functions for degenerate elliptic operators

2020 ◽  
Vol 13 (1) ◽  
pp. 75-113 ◽  
Author(s):  
Li Chen ◽  
José María Martell ◽  
Cruz Prisuelos-Arribas
Keyword(s):  
Divergence Form ◽  
Elliptic Operators ◽  
Square Functions ◽  
Degenerate Elliptic Operators ◽  
Weighted Estimates ◽  
Poisson Semigroup ◽  
Bounded Complex ◽  
Degenerate Elliptic ◽  
Complex Valued ◽  
Kato Square Root Problem

AbstractThe aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)}, where {w\in A_{2}} and A is an {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the {L^{2}(w)}-Kato square root problem obtaining that {\sqrt{L_{w}}} is equivalent to the gradient on {L^{2}(w)}. The same authors in collaboration with the second named author of this paper studied the {L^{p}(w)}-boundedness of operators that are naturally associated with {L_{w}}, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in {L^{p}(v\,dw)} for {v\in A_{\infty}(w)}), and in particular a class of “degeneracy” weights w was found in such a way that the classical {L^{2}}-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on {L^{p}(w)} and on {L^{p}(v\,dw)}, with {v\in A_{\infty}(w)}, of the conical square functions that one can construct using the heat or Poisson semigroup associated with {L_{w}}. As a consequence of our methods, we find a class of degeneracy weights w for which {L^{2}}-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with {L_{w}}.

scholarly journals Multiple-weighted norm inequalities for maximal multi-linear singular integrals with non-smooth kernels

2011 ◽  
Vol 141 (4) ◽  
pp. 755-775 ◽  
Author(s):  
Loukas Grafakos ◽  
Liguang Liu ◽  
Dachun Yang
Keyword(s):  
Singular Integrals ◽  
Weighted Norm ◽  
Weighted Norm Inequalities ◽  
Norm Inequalities ◽  
Weighted Estimates ◽  
New Class ◽  
Littlewood Maximal Function ◽  
Smooth Kernels ◽  
Multiple Weights

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al. This class of operators extends the class of multi-linear Calderón-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar's inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced.

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