ON SECOND-ORDER ALMOST-PERIODIC ELLIPTIC OPERATORS

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.

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

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870000094x ◽

1997 ◽

Vol 63 (3) ◽

pp. 297-363 ◽

Cited By ~ 1

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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On the Stepanov-almost periodic solution of a second-order operator differential equation

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500015546 ◽

1975 ◽

Vol 19 (3) ◽

pp. 261-263 ◽

Cited By ~ 5

Author(s):

Aribindi Satyanarayan Rao

Keyword(s):

Differential Equation ◽

Banach Space ◽

Periodic Solution ◽

Almost Periodic Solution ◽

Second Order ◽

Almost Periodicity ◽

Operator Differential Equation ◽

Almost Periodic ◽

Order Operator ◽

Image Position

Suppose X is a Banach space and J is the interval −∞<t<∞. For 1 ≦ p<∞, a function is said to be Stepanov-bounded or Sp-bounded on J if(for the definitions of almost periodicity and Sp-almost periodicity, see Amerio-Prouse (1, pp. 3 and 77).

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A complete asymptotic expansion of the spectral function for second-order elliptic operators inRn

Russian Mathematical Surveys ◽

10.1070/rm1983v038n01abeh003411 ◽

1983 ◽

Vol 38 (1) ◽

pp. 213-214

Author(s):

G S Popov ◽

M A Shubin

Keyword(s):

Asymptotic Expansion ◽

Spectral Function ◽

Elliptic Operators ◽

Second Order ◽

Complete Asymptotic Expansion ◽

Second Order Elliptic Operators

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Commutators with Lipschitz Functions and Nonintegral Operators

Journal of Applied Mathematics ◽

10.1155/2013/178961 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 1

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.

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