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.

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.

ON SECOND-ORDER ALMOST-PERIODIC ELLIPTIC OPERATORS

2001 ◽  
Vol 63 (3) ◽  
pp. 735-753 ◽  
Author(s):  
N. DUNGEY ◽  
A. F. M. TER ELST ◽  
DEREK W. ROBINSON
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Behaviour ◽  
Large Time ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
Hölder Continuous ◽  
Leading Term

The paper considers second-order, strongly elliptic, operators H with complex almost-periodic coefficients in divergence form on Rd. First, it is proved that the corresponding heat kernel is Hölder continuous and Gaussian bounds are derived with the correct small and large time asymptotic behaviour on the kernel and its Hölder derivatives. Secondly, it is established that the kernel has a variety of properties of almost-periodicity. Thirdly, it is demonstrated that the kernel of the homogenization Ĥ of H is the leading term in the asymptotic expansion of t [map ] Kt.

scholarly journals High order divergence-form elliptic operators on Lie groups

1997 ◽  
Vol 55 (2) ◽  
pp. 335-348 ◽  
Author(s):  
A. F. M. ter Elst ◽  
Derek W. Robinson
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Divergence Form ◽  
Elliptic Operators ◽  
High Order ◽  
Hölder Continuous ◽  
Gaussian Bounds ◽  
Straightforward Proof

We give a straightforward proof that divergence-form elliptic operators of ordermon ad-dimensional Lie group withm≥dhave Hölder continuous kernels satisfying Gaussian bounds.

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.

scholarly journals Commutators with Lipschitz Functions and Nonintegral Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Peizhu Xie ◽  
Ruming Gong
Keyword(s):  
Integral Representation ◽  
Fractional Power ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Lipschitz Functions ◽  
Positive Operators ◽  
Riesz Transforms ◽  
Spectral Multipliers ◽  
Size Estimates

LetTbe a singular nonintegral operator; that is, it does not have an integral representation by a kernel with size estimates, even rough. In this paper, we consider the boundedness of commutators withTand Lipschitz functions. Applications include spectral multipliers of self-adjoint, positive operators, Riesz transforms of second-order divergence form operators, and fractional power of elliptic operators.

scholarly journals Second Order Perturbations of Elliptic Elements with Respect to the Initial Ones

1999 ◽  
Vol 172 ◽  
pp. 453-454
Author(s):  
F.J. Marco Castillo ◽  
M.J. Martínez Usó ◽  
J.A. López Ortí
Keyword(s):  
Quadratic Form ◽  
Second Order ◽  
Initial Instant ◽  
Orbital Elements ◽  
Partial Derivatives ◽  
Orbital Parameters ◽  
The Matrix ◽  
The Individual ◽  
Derivatives Of

AbstractThe following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged: 1.To be sure that the most suitable quadratic form was to be considered.2.To provide a wider vision of the behavior of the different orbital parameters in time.Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.

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.

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