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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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Cited by 23 (6 self)
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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
Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds, preprint Arxxiv: math.AP/0701515
"... Abstract. Let M ◦ be a complete noncompact manifold and g an asymptotically conic metric on M ◦ , in the sense that M ◦ compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. A special case of particular interest is that of asymptotically Euclidean manifold ..."
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Cited by 11 (2 self)
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Abstract. Let M ◦ be a complete noncompact manifold and g an asymptotically conic metric on M ◦ , in the sense that M ◦ compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. A special case of particular interest is that of asymptotically Euclidean manifolds, where ∂M = S n−1 and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel (P + k 2) −1 and Riesz transform of the operator P = ∆g + V, where ∆g is the positive Laplacian associated to g and V is a real potential function V that is smooth on M and vanishes to some finite order at the boundary. In the first paper in this series we made the assumption that n ≥ 3 and that P has neither zero modes nor a zeroresonance and showed (i) that the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary on a blown up version of M 2 × [0, k0], and (ii) the Riesz transform of P is bounded on L p (M ◦ ) for 1 < p < n, and that this range is optimal unless V ≡ 0 and M ◦ has only one end. In the present paper, we perform a similar analysis assuming again n ≥ 3 but allowing zero modes and zeroresonances. We show that once again that (unless n = 4 and there is a zeroresonance) the resolvent kernel is polyhomogeneous on the same space and compute its leading asymptotics. This generalizes results of JensenKato and Murata to the variable coefficient setting. We also find the precise range of p for which the Riesz transform (suitably defined) of P is bounded on L p (M) when zero modes (but not resonances, which make the Riesz transform undefined) are present. Generically the Riesz transform is bounded for p precisely in the range (n/(n − 2), n/3), with a bigger range possible if the zero modes have extra decay at infinity. 1.
Bounds of Riesz transforms on L p spaces for second order elliptic operators
 Ann. Inst. Fourier
"... Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Rie ..."
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Cited by 10 (1 self)
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Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L) −1/2 on the L p space. As an application, for 1 < p < 3 + ε, we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied. 1.
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier, Grenoble 57 no 6
, 2007
"... The paper concerns the magnetic Schrödinger operator H(a, V) = ..."
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Cited by 9 (3 self)
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The paper concerns the magnetic Schrödinger operator H(a, V) =
Riesz transform and L p cohomology for manifolds with Euclidean ends
 Duke Math. J
"... Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n ..."
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Cited by 8 (2 self)
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Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n and unbounded for p ≥ n if there is more than one end. It follows from known results that in such a case the Riesz transform on M is bounded for 1 < p ≤ 2 and unbounded for p> n; the result is new for 2 < p ≤ n. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in L p for some p> 2 for a more general class of manifolds. Assume that M is a ndimensional complete manifold satisfying the Nash inequality and with an O(r n) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on L p for some p> 2 implies a Hodgede Rham interpretation of the L p cohomology in degree 1, and that the map from L 2 to L p cohomology in this degree is injective. 1.
SOME GRADIENT ESTIMATES FOR THE HEAT EQUATION ON DOMAINS AND FOR AN EQUATION BY PERELMAN
, 2006
"... Abstract. In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local LiYau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bo ..."
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Cited by 7 (0 self)
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Abstract. In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local LiYau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound. Contents
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
Hypoelliptic heat kernel inequalities on Lie groups
, 2005
"... This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associate ..."
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Cited by 6 (1 self)
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This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature ” takes on the value − ∞ at points of degeneracy of the semiRiemannian metric associated to the operator. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are invariant under left translation. In particular, “L ptype ” gradient estimates hold for p ∈ (1, ∞), and the p = 2 gradient estimate
Decay Preserving Operators and stability of the essential spectrum, preprint math.SP/0411489, to appear in Journal of Operator Theory
"... We establish some criteria for the stability of the essential spectrum for unbounded operators acting in Banach modules. The applications cover operators acting on sections of vector fiber bundles over nonsmooth manifolds or locally compact abelian groups, in particular differential operators of an ..."
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Cited by 6 (5 self)
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We establish some criteria for the stability of the essential spectrum for unbounded operators acting in Banach modules. The applications cover operators acting on sections of vector fiber bundles over nonsmooth manifolds or locally compact abelian groups, in particular differential operators of any order with complex measurable coefficients on R n, singular Dirac operators and LaplaceBeltrami operators on Riemannian manifolds with measurable metrics. 1
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)