Weighted estimates for bilinear square functions with non-smooth kernels and commutators

2020 ◽  
Vol 15 (1) ◽  
pp. 1-20
Author(s):  
Rui Bu ◽  
Zunwei Fu ◽  
Yandan Zhang
Keyword(s):  
Square Functions ◽  
Weighted Estimates ◽  
Smooth Kernels

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 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 Estimates for Vector-Valued Multilinear Singular Integrals with Non-Smooth Kernels and Its Commutators

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Dongxiang Chen ◽  
Dan Zou ◽  
Suzhen Mao
Keyword(s):  
Singular Integrals ◽  
Weighted Norm ◽  
Multilinear Commutator ◽  
Weighted Estimates ◽  
Bmo Functions ◽  
Multilinear Singular Integral Operator ◽  
Smooth Kernels ◽  
New Variant ◽  
Vector Valued ◽  
Multilinear Singular Integrals

This note concerns multiple weighted inequalities for vector-valued multilinear singular integral operator with nonsmooth kernel and its corresponding commutators containing multilinear commutator and iterated commutator generated by the vector-valued multilinear operator and BMO functions. By the weighted estimates for a class of new variant maximal and sharp maximal functions, the multiple weighted norm inequalities for such operators are obtained.

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