scholarly journals Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the Lp Dirichlet Problem

2019 ◽  
Vol 35 (6) ◽  
pp. 749-770
Author(s):  
Martin Dindoš ◽  
Jill Pipher
Keyword(s):  
Perturbation Theory ◽  
Dirichlet Problem ◽  
Elliptic Operators ◽  
Second Order ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

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 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.

The Dirichlet Problem for Second Order Elliptic Operators on Unbounded Domains

1985 ◽  
Vol 19 (2-3) ◽  
pp. 201-216 ◽  
Author(s):  
W. Walter ◽  
Rainer Janben ◽  
V. Besov
Keyword(s):  
Dirichlet Problem ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Second Order ◽  
Second Order Elliptic Operators

Some results on second-order elliptic operators with polynomially growing coefficients in L-spaces

2021 ◽  
pp. 125209
Author(s):  
Sallah Eddine Boutiah ◽  
Loredana Caso ◽  
Federica Gregorio ◽  
Cristian Tacelli
Keyword(s):  
Elliptic Operators ◽  
Second Order ◽  
Second Order Elliptic Operators
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