scholarly journals $$L^p$$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

2022 ◽  
Author(s):  
Steve Hofmann ◽  
Linhan Li ◽  
Svitlana Mayboroda ◽  
Jill Pipher
Keyword(s):  
Elliptic Operators ◽  
Symmetric Part ◽  
Square Functions ◽  
Square Roots

scholarly journals The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

2021 ◽  
Author(s):  
Steve Hofmann ◽  
Linhan Li ◽  
Svitlana Mayboroda ◽  
Jill Pipher
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Operators ◽  
Symmetric Part

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 Square roots of elliptic operators

1985 ◽  
Vol 61 (3) ◽  
pp. 307-327 ◽  
Author(s):  
Alan McIntosh
Keyword(s):  
Elliptic Operators ◽  
Square Roots

scholarly journals Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Xiongtao Wu ◽  
Wenyu Tao ◽  
Yanping Chen ◽  
Kai Zhu
Keyword(s):  
Elliptic Operator ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Square Function ◽  
Fractional Differentiation ◽  
Square Functions ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

Let L=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators.

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