scholarly journals On the $$\mathrm {L}^p$$-theory for second-order elliptic operators in divergence form with complex coefficients

2021 ◽  
Author(s):  
A. F. M. ter Elst ◽  
R. Haller-Dintelmann ◽  
J. Rehberg ◽  
P. Tolksdorf
Keyword(s):  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

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.

Commutators with fractional differentiation for second-order elliptic operators on ℝn

2019 ◽  
Vol 22 (02) ◽  
pp. 1950010
Author(s):  
Yanping Chen ◽  
Qingquan Deng ◽  
Yong Ding
Keyword(s):  
Differential Operators ◽  
Divergence Form ◽  
Second Order ◽  
Bounded Mean Oscillation ◽  
Fractional Differentiation ◽  
Mean Oscillation ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

Let [Formula: see text] be a second-order divergence form elliptic operator and [Formula: see text] an accretive, [Formula: see text] matrix with bounded measurable complex coefficients in [Formula: see text] In this paper, we establish [Formula: see text] theory for the commutators generated by the fractional differential operators related to [Formula: see text] and bounded mean oscillation (BMO)–Sobolev functions.

Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces

2018 ◽  
Vol 30 (3) ◽  
pp. 617-629 ◽  
Author(s):  
Yanping Chen ◽  
Yong Ding
Keyword(s):  
Fractional Integral ◽  
Uniform Boundedness ◽  
Morrey Spaces ◽  
Elliptic Operators ◽  
Second Order ◽  
Fractional Integrals ◽  
Riesz Potentials ◽  
Heat Operator ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

AbstractLet {L=-\operatorname{div}(A\nabla)} be a second-order divergence form elliptic operator and let A be an accretive, {n\times n} matrix with bounded measurable complex coefficients in {{\mathbb{R}}^{n}}. Let {L^{-\frac{\alpha}{2}}} be the fractional integral associated to L for {0<\alpha<n}. For {b\in L_{\mathrm{loc}}({\mathbb{R}}^{n})} and {k\in{\mathbb{N}}}, the k-th order commutator of b and {L^{-\frac{\alpha}{2}}} is given by(L^{-\frac{\alpha}{2}})_{b,k}f(x)=L^{-\frac{\alpha}{2}}((b(x)-b)^{k}f)(x).In the paper, we mainly show that if {b\in\mathrm{BMO}({\mathbb{R}}^{n})}, {0<\lambda<n} and {0<\alpha<n-\lambda}, then {(L^{-\frac{\alpha}{2}})_{b,k}} is bounded from {L^{p,\lambda}} to {L^{q,\lambda}} for {p_{-}(L)<p<q<p_{+}(L)\frac{n-\lambda}{n}} and {\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n-\lambda}}, where {p_{-}(L)} and {p_{+}(L)} are the two critical exponents for the {L^{p}} uniform boundedness of the semigroup {\{e^{-tL}\}_{t>0}}. Also, we establish the boundedness of the commutator of the fractional integral with Lipschitz function on Morrey spaces. The results encompass what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator.

scholarly journals Second-order strongly elliptic operators on Lie groups with Hölder continuous coefficients

1997 ◽  
Vol 63 (3) ◽  
pp. 297-363 ◽  
Author(s):  
A. F. M. Ter Elst ◽  
Derek W. Robinson
Keyword(s):  
Quadratic Form ◽  
Lie Group ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Hölder Continuous ◽  
The Matrix ◽  
Algebraic Basis ◽  
Complex Coefficients ◽  
Connected Lie Group

AbstractLet G be a connected Lie group with Lie algebra g and a1, …, ad an algebraic basis of g. Further let Ai denote the generators of left translations, acting on the Lp-spaces Lp(G; dg) formed with left Haar measure dg, in the directions ai. We consider second-order operators in divergence form corresponding to a quadratic form with complex coefficients, bounded Hölder continuous principal coefficients cij and lower order coefficients ci, c′ii, c0 ∈ L∞ such that the matrix C= (cij) of principal coefficients satisfies the subellipticity condition uniformly over G.We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel and smoothness of the domain of powers of H on the Lρ-spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives.Similar theorems are proved for strongly elliptic operators in non-divergence form for which the principal coefficients are at least once differentiable.

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