Results 1  10
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66
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 44 (13 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 31 (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.
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 16 (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 . . . . . . . . . . . . . . . . .
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 14 (9 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
Geometric implications of the Poincaré inequality
 Results Math
"... Abstract. The purpose of this work is to prove the following result: If a doubling metric measure space supports a weak (1, p)–Poincare ́ inequality with p sufficiently small, then annuli are almost quasiconvex. We also obtain estimates for the Hausdorff s–content and the diameter of the spheres. ..."
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Cited by 13 (3 self)
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Abstract. The purpose of this work is to prove the following result: If a doubling metric measure space supports a weak (1, p)–Poincare ́ inequality with p sufficiently small, then annuli are almost quasiconvex. We also obtain estimates for the Hausdorff s–content and the diameter of the spheres.
An Elementary Proof Of Sharp Sobolev Embeddings
, 2000
"... . We present an elementary unified and selfcontained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Br#zis and Wainger. 1. Prologue Let\Omega be an open subset of R n , where n 2, let 1 p ! 1 and ..."
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Cited by 11 (1 self)
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. We present an elementary unified and selfcontained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Br#zis and Wainger. 1. Prologue Let\Omega be an open subset of R n , where n 2, let 1 p ! 1 and let W 1;p (\Omega\Gamma be the Sobolev space, that is, the set of all functions in L p(\Omega\Gamma8 whose distributional derivatives of the first order belong to L p(\Omega\Gamma9 too. If p = n we assume that j\Omega j ! 1. We define W 1;p 0 (\Omega\Gamma as the closure of C 1 0 (\Omega\Gamma in W 1;p(\Omega\Gamma3 We denote p = np n \Gamma p ; 1 p ! n: The classical Sobolev theorem [16] asserts that W 1;p 0 (\Omega\Gamma ,! L p when 1 ! p ! n: (1.1) (As usual, ,! stands for a continuous embedding.) Although p tends to infinity as p ! n\Gamma, the space W 1;n 0 (\Omega\Gamma contains unbounded functions. Instead of an embedding into L 1 (\Omega\Gamma1 one h...