Results 1  10
of
31
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
Abstract

Cited by 51 (5 self)
 Add to MetaCart
We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Risk communication
 Proceedings of the national conference on risk communication, Conservation Foundation,Washington, DC
, 1987
"... We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a fo ..."
Abstract

Cited by 37 (1 self)
 Add to MetaCart
(Show Context)
We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x) 1.x‘.:Ta 1991 Academic Press, Inc. 1.
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 ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
(Show Context)
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.
Upper bounds for eigenvalues of the discrete and continuous Laplace operators
 Adv. Math
, 1996
"... In this paper, we are concerned with upper bounds of eigenvalues of Laplace operator on compact Riemannian manifolds and finite graphs. While on the former the Laplace operator is generated by the Riemannian metric, on the latter it reflects combinatorial structure of a graph. Respectively, eigenval ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
(Show Context)
In this paper, we are concerned with upper bounds of eigenvalues of Laplace operator on compact Riemannian manifolds and finite graphs. While on the former the Laplace operator is generated by the Riemannian metric, on the latter it reflects combinatorial structure of a graph. Respectively, eigenvalues have many applications in geometry as well
HARDY SPACES ASSOCIATED TO NONNEGATIVE SELFADJOINT OPERATORS SATISFYING DAVIESGAFFNEY ESTIMATES
"... Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characte ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a nonnegative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, (X) for p> 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L(X) spaces by the complex method. we define Hardy spaces H p L The authors gratefully acknowledge support from NSF as follows: S. Hofmann (DMS
A DIRECT LEBEAUROBBIANO STRATEGY FOR THE OBSERVABILITY OF HEATLIKE SEMIGROUPS
"... Dedicated to David L. Russell on the occasion of his 70th birthday Abstract. This paper generalizes and simplifies abstract results of Miller and Seidman on the cost of fast control/observation. It deduces finalobservability of an evolution semigroup from a spectral inequality, i.e. some stationary ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
Dedicated to David L. Russell on the occasion of his 70th birthday Abstract. This paper generalizes and simplifies abstract results of Miller and Seidman on the cost of fast control/observation. It deduces finalobservability of an evolution semigroup from a spectral inequality, i.e. some stationary observability property on some spaces associated to the generator, e.g. spectral subspaces when the semigroup has an integral representation via spectral measures. Contrary to the original LebeauRobbiano strategy, it does not have recourse to nullcontrollability and it yields the optimal bound of the cost when applied to the heat equation, i.e. c0 exp(c/T), or to the heat diffusion in potential wells observed from cones, i.e. c0 exp(c/T β) with optimal β. It also yields simple upper bounds for the cost rate c in terms of the spectral rate. This paper also gives geometric lower bounds on the spectral and cost rates for heat, diffusion and GinzburgLandau semigroups, including on noncompact Riemannian manifolds, based on L 2 Gaussian estimates.
Gaussian heat kernel upper bounds via the PhragmnLindelf theorem
 Proc. Lond. Math. Soc
"... Abstract. We prove that in presence of L 2 Gaussian estimates, socalled DaviesGaffney estimates, ondiagonal upper bounds imply precise offdiagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Contents ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Abstract. We prove that in presence of L 2 Gaussian estimates, socalled DaviesGaffney estimates, ondiagonal upper bounds imply precise offdiagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Contents
OrliczHardy Spaces Associated with Operators Satisfying DaviesGaffney Estimates
, 903
"... Abstract. Let X be a metric space with doubling measure, L be a nonnegative selfadjoint operator in L2 (X) satisfying the DaviesGaffney estimate, ω be a concave function on (0, ∞) of strictly lower type pω ∈ (0, 1] and ρ(t) = t−1 /ω−1(t−1) for all t ∈ (0, ∞). The authors introduce the OrliczHardy ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract. Let X be a metric space with doubling measure, L be a nonnegative selfadjoint operator in L2 (X) satisfying the DaviesGaffney estimate, ω be a concave function on (0, ∞) of strictly lower type pω ∈ (0, 1] and ρ(t) = t−1 /ω−1(t−1) for all t ∈ (0, ∞). The authors introduce the OrliczHardy space Hω,L(X) via the Lusin area function associated to the heat semigroup, and the BMOtype space BMOρ,L(X). The authors then establish the duality between Hω,L(X) and BMOρ,L(X); as a corollary, the authors obtain the ρCarleson measure characterization of the space BMOρ,L(X). Characterizations of Hω,L(X), including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. As applications, the authors obtain the characterizations of the Hardy spaces H p L (Rn) associated to the Schrödinger operator L = − ∆ + V, where V ∈ L1 loc (Rn) is a nonnegative potential and p ∈ (0, 1], in terms of the Lusinarea functions, the nontangential maximal functions, the radial maximal functions, the atoms and the molecules. Finally, the authors show that the Riesz transform ∇L−1/2 is bounded from H p L (Rn) to Lp (Rn) for all p ∈ (0, 1] and from H p L (Rn) to the classical Hardy space Hp (Rn) for p ∈ ( n n+1, 1]. All these results are new even when ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1). 1