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.

Homogeneous Calderón–Zygmund estimates for a class of second-order elliptic operators

2014 ◽  
Vol 17 (01) ◽  
pp. 1450017 ◽  
Author(s):  
G. P. Galdi ◽  
G. Metafune ◽  
C. Spina ◽  
C. Tacelli
Keyword(s):  
Elliptic Operator ◽  
Unique Solvability ◽  
Elliptic Operators ◽  
Second Order ◽  
Poisson Problem ◽  
Second Order Elliptic Operators ◽  
Uniformly Continuous

We prove unique solvability and corresponding homogeneous Lp estimates for the Poisson problem associated to the uniformly elliptic operator [Formula: see text], provided the coefficients are bounded and uniformly continuous, and admit a (non-zero) limit as |x| goes to infinity. Some important consequences are also derived.

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 Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators

2002 ◽  
Vol 165 ◽  
pp. 123-158 ◽  
Author(s):  
Alano Ancona
Keyword(s):  
Harmonic Function ◽  
Elliptic Operator ◽  
Critical Points ◽  
Harmonic Functions ◽  
Elliptic Operators ◽  
Second Order ◽  
Small Perturbations ◽  
Order Elliptic Operator ◽  
Symmetry Assumption ◽  
Second Order Elliptic Operators

Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

Sign in / Sign up

Export Citation Format

Share Document