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Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
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Cited by 71 (4 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
Heat kernels on metricmeasure spaces and an application to semilinear elliptic equations
 Trans. Amer. Math. Soc
, 2003
"... Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ) ..."
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Cited by 34 (13 self)
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Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ). Namely, α is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M. Moreover, the parameters α and β are related by the inequalities 2 ≤ β ≤ α +1. We prove also the embedding theorems for the space W β/2,2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form −Lu + f(x, u) =g(x), where L is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in Rn. 1.
UNIVERSAL BOUNDS FOR TRACES OF THE DIRICHLET LAPLACE OPERATOR
, 909
"... Abstract. We derive upper bounds for the trace of the heat kernel Z(t) = ..."
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Cited by 5 (2 self)
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Abstract. We derive upper bounds for the trace of the heat kernel Z(t) =
Lemma 1. Let A ∈B(H). The following are equivalent. (i) The limit
"... A CPsemigroup (or quantum dynamical semigroup) is a semigroup φ = {φt: t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t ≥ 0 and is continuous in the time parameter t the natural sense. Let D be the natural domain of th ..."
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A CPsemigroup (or quantum dynamical semigroup) is a semigroup φ = {φt: t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t ≥ 0 and is continuous in the time parameter t the natural sense. Let D be the natural domain of the generator L of φ, φt = exp tL, t ≥ 0. Since the maps φt need not be multiplicative D is typically an operator space, but not an algebra. However, in this note we show that the set of operators A = {A ∈D: A ∗ A ∈D,AA ∗ ∈D} is a ∗subalgebra of B(H), indeed A is the largest selfadjoint algebra contained in D. Examples are described for which the domain algebra A is, and is not, strongly dense in B(H). 1. Basic properties of A. Let φ = {φt: t ≥ 0} be a CPsemigroup as defined in the abstract. We first recall four characterizations of the domain of the generator of φ.
Supported by the Austrian Federal Ministry of Education, Science and Culture
"... Abstract. We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set Ω ⊂ R d, d ≥ 2. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on Z(t) in domains of infinite volume. For domains of fi ..."
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Abstract. We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set Ω ⊂ R d, d ≥ 2. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on Z(t) in domains of infinite volume. For domains of finite volume the bound on Z(t) decays exponentially as t tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of Z(t). To prove the result we employ refined BerezinLiYau inequalities for eigenvalue means. 1. Introduction and
t→0+
, 2002
"... A CPsemigroup (or quantum dynamical semigroup) is a semigroup φ = {φt: t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t ≥ 0 and is continuous in the time parameter t the natural sense. Let D be the natural domain of ..."
Abstract
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A CPsemigroup (or quantum dynamical semigroup) is a semigroup φ = {φt: t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t ≥ 0 and is continuous in the time parameter t the natural sense. Let D be the natural domain of the generator L of φ, φt = exp tL, t ≥ 0. Since the maps φt need not be multiplicative D is typically an operator space, but not an algebra. However, in this note we show that the set of operators A = {A ∈ D: A∗A ∈ D, AA ∗ ∈ D} is a ∗subalgebra of B(H), indeed A is the largest selfadjoint algebra contained in D. Examples are described for which the domain algebra A is, and is not, strongly dense in B(H). 1. Basic properties of A. Let φ = {φt: t ≥ 0} be a CPsemigroup as defined in the abstract. We first recall four characterizations of the domain of the generator of φ. Lemma 1. Let A ∈ B(H). The following are equivalent.
(i) The limit
, 2000
"... Abstract. A CPsemigroup (or quantum dynamical semigroup) is a semigroup φ = {φt: t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t and is continuous in the natural sense. Let D be the natural domain of the generator L o ..."
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Abstract. A CPsemigroup (or quantum dynamical semigroup) is a semigroup φ = {φt: t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t and is continuous in the natural sense. Let D be the natural domain of the generator L of φ, φt = exp tL. Since the maps φt need not be multiplicative D is typically an operator space, but not an algebra. However, we show that the set of operators A = {A ∈ D: A ∗ A ∈ D, AA ∗ ∈ D} is a ∗subalgebra of B(H), indeed A is the largest selfadjoint algebra contained in D. Because A is a ∗algebra one may consider its ∗bimodule of noncommutative 2forms Ω 2 (A) = Ω 1 (A) ⊗A Ω 1 (A), and any linear mapping L: A → B(H) has a symbol σL: Ω 2 (A) → B(H), defined as a linear map by σL(a dx dy) = aL(xy) − axL(y) − aL(x)y + axL(1)y, a, x, y ∈ A. The symbol is a homomorphism of Abimodules for any ∗algebra A ⊆ B(H) and any linear map L: A → B(H). When L is the generator of a CPsemigroup with domain algebra A above, we show that the symbol is negative in that σL(ω ∗ ω) ≤ 0 for every ω ∈ Ω 1 (A) (−σL is in fact completely positive). Examples are given for which the domain algebra A is, and is not, strongly dense in B(H). We also relate the generator of a CPsemigroup to its commutative paradigm, the Laplacian of a Riemannian manifold. 1. Basic properties of A. Let φ = {φt: t ≥ 0} be a CPsemigroup as defined in the abstract. We first recall four characterizations of the domain of the generator of φ.