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Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
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 ..."
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Cited by 7 (0 self)
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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
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 ..."
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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
Fractional Poincaré inequality for general measures
"... Abstract. We prove a fractional version of Poincaré inequalities in the context of R n endowed with a fairly general measure. Namely we prove a control of an L 2 norm by a non local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is t ..."
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Cited by 6 (3 self)
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Abstract. We prove a fractional version of Poincaré inequalities in the context of R n endowed with a fairly general measure. Namely we prove a control of an L 2 norm by a non local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the OrnsteinUhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.
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 ..."
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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.
Interpolation of Sobolev spaces, LittlewoodPaley inequalities and Riesz transforms on graphs
 PUBLICACIONS MATEMATIQUES
"... Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P) ..."
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Cited by 5 (2 self)
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Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P)
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 ..."
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Cited by 4 (3 self)
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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
Stochastically Incomplete Manifolds and Graphs
"... Abstract. We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to ..."
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Abstract. We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of stochastically incomplete graphs of polynomial volume growth.