Results 21  30
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241
Riesz transform, Gaussian bounds and the method of wave equation
 Math. Z
"... Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We al ..."
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Cited by 20 (1 self)
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Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the LaplaceBeltrami operator on Riemannian manifolds for p> 2. 1.
Operator Kernel Estimates For Functions Of Generalized Schrödinger Operators
 Proc. Amer. Math. Soc
, 2001
"... We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of Mathematical Physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave ..."
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Cited by 20 (8 self)
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We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of Mathematical Physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved CombesThomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.
Hypercontractivity for perturbed diffusion semigroups
 ANN. FAC. DES SC. DE TOULOUSE
, 2005
"... µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary conditio ..."
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Cited by 20 (14 self)
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µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied.
The logrithmic Sobolev inequality along the Ricci flow, arXiv:0707.2424v4
"... 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow ..."
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Cited by 18 (2 self)
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2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow
Várilly, “Dixmier traces on noncompact isospectral deformations
 J. Funct. Anal
"... We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of fu ..."
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Cited by 18 (8 self)
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We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of R l, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Sharp heat kernel estimates for relativistic stable processes in open sets
"... In this paper, we establish sharp twosided estimates for the transition densities of relativistic stable processes (or equivalently, for the heat kernels of the operators m − (m 2/α − ∆) α/2) in C 1,1 open sets. The estimates are uniform in m ∈ (0,M] for each fixed M> 0. Letting m ↓ 0, the estimate ..."
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Cited by 17 (14 self)
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In this paper, we establish sharp twosided estimates for the transition densities of relativistic stable processes (or equivalently, for the heat kernels of the operators m − (m 2/α − ∆) α/2) in C 1,1 open sets. The estimates are uniform in m ∈ (0,M] for each fixed M> 0. Letting m ↓ 0, the estimates given in this paper recover the Dirichlet heat kernel estimates for −(−∆) α/2 in C 1,1open sets obtained in [9]. Sharp twosided estimates are also obtained for Green functions of relativistic stable processes in halfspacelike C 1,1 open sets and bounded C 1,1 open sets.
Sharp bounds on the density, Green function and jumping function of subordinate killed BM
 PROBAB. THEORY RELAT. FIELDS
, 2004
"... Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2stable subordinator gives rise to a process Zt whose infinitesimal generator is −(−�D) α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green ..."
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Cited by 15 (12 self)
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Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2stable subordinator gives rise to a process Zt whose infinitesimal generator is −(−�D) α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Zt when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.
Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains
, 1998
"... For a symmetric ffstable process X on R n with 0 ! ff ! 2, n 2 and a domain D ae R n , let L D be the infinitesimal generator of the subprocess of X killed upon leaving D. For a Kato class function q, it is shown that L D + q is intrinsic ultracontractive on a Holder domain D of order 0. ..."
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Cited by 15 (6 self)
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For a symmetric ffstable process X on R n with 0 ! ff ! 2, n 2 and a domain D ae R n , let L D be the infinitesimal generator of the subprocess of X killed upon leaving D. For a Kato class function q, it is shown that L D + q is intrinsic ultracontractive on a Holder domain D of order 0. This is then used to establish the conditional gauge theorem for X on bounded Lipschitz domains in R n . It is also shown that the conditional lifetimes for symmetric stable process in a Holder domain of order 0 are uniformly bounded. Keywords and phrases: Symmetric stable processes, FeynmanKac semigroup, conditional gauge theorem, logarithmic Sobolev inequality, intrinsic ultracontractivity. Running Title: Conditional Gauge Theorem The research of this author is supported in part by NSA Grant MDA9049810044 1 Introduction A symmetric ffstable process X on R n is a L'evy process whose transition density p(t; x \Gamma y) relative to Lebesgue measure is uniquely determined by ...
Explicit Constants for Rellich Inequalities in ...
, 1997
"... Introduction Let\Omega be a bounded region in a complete Riemannian manifold with smooth boundary @ Let C 1 (\Omega ); C 1 0 (\Omega\Gamma and C 1 c (\Omega\Gamma denote respectively the space of smooth functions on\Omega , the subspace consisting of such functions which vanish on @ and the su ..."
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Cited by 15 (2 self)
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Introduction Let\Omega be a bounded region in a complete Riemannian manifold with smooth boundary @ Let C 1 (\Omega ); C 1 0 (\Omega\Gamma and C 1 c (\Omega\Gamma denote respectively the space of smooth functions on\Omega , the subspace consisting of such functions which vanish on @ and the subspace of such functions which vanish in a neighbourhood of @ \Omega\Gamma For those whose main interest is in spectral theory in Euclidean space we mention that our main results are also new in that context. We investigate the existence and explicit determination of constants c and weights X and Y on\Omega such that the Rellich inequality Z \Omega Xjuj p c Z \Omega Y