Results 1  10
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89
Best constants for GagliardoNirenberg inequalities and applications to nonlinear diffusions
, 2001
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Functional inequalities for empty essential spectrum
 J. Funct. Anal
, 2000
"... In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corre ..."
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Cited by 67 (14 self)
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In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for superPoincare ́ inequalities is studied, the resulting estimates extend previous ones for Poincare ́ and logSobolev inequalities.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Manifolds and Graphs With Slow Heat Kernel Decay
 Invent. Math
, 1999
"... We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."
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Cited by 35 (4 self)
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We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics
 In preparation
"... We study the effective largescale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the correcto ..."
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Cited by 32 (18 self)
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We study the effective largescale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinitedimensional space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w. r. t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i. e. for the “random environment as seen from a random walker”). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension d> 2) and optimal estimates for regularized versions of the corrector (in dimensions d ≥ 2). We also give a selfcontained proof for a new estimate on the gradient of the parabolic, variablecoefficient Green’s function, which is a crucial analytic ingredient in our method. As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative
Harnack inequality and hyperbolicity for subelliptic pLaplacians with applications to Picard type theorems
, 2000
"... Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . ..."
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Cited by 20 (4 self)
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Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The pLaplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The nonsmooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 pparabolicity and phyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and pparabolicity . . . . . . . . . . . . . . . . .
Sobolev Algebras on Lie groups and Riemannian manifolds
 Amer. J. Math
"... Abstract. We prove that on any connected unimodular Lie group G, the space Lp ..."
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Cited by 19 (5 self)
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Abstract. We prove that on any connected unimodular Lie group G, the space Lp
Riesz transform and Lp 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, Rn \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 Lp(M) → Lp(M;T ∗M) for 1 < p < n ..."
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Cited by 17 (5 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, Rn \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 Lp(M) → Lp(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 Lp 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(rn) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on Lp for some p> 2 implies a Hodgede Rham interpretation of the Lp cohomology in degree 1, and that the map from L2 to Lp cohomology in this degree is injective. 1.