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37
Best constants for GagliardoNirenberg inequalities and applications to nonlinear diffusions
"... In this paper, we find optimal constants of a special class of GagliardoNirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of ..."
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Cited by 62 (16 self)
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In this paper, we find optimal constants of a special class of GagliardoNirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diffusion equations.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
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 24 (2 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 8 (4 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
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.
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 6 (0 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...
Ultracontractive bounds on HamiltonJacobi solutions
 Bull. Sc. math
, 2002
"... Following the equivalence between logarithmic Sobolev inequality, hypercontractivity of the heat semigroup showed by Gross and hypercontractivity of HamiltonJacobi equations, we prove, like the Varopoulos theorem, the equivalence between Euclideantype Sobolev inequality and an ultracontractive con ..."
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Cited by 6 (4 self)
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Following the equivalence between logarithmic Sobolev inequality, hypercontractivity of the heat semigroup showed by Gross and hypercontractivity of HamiltonJacobi equations, we prove, like the Varopoulos theorem, the equivalence between Euclideantype Sobolev inequality and an ultracontractive control of the HamiltonJacobi equations. We obtain also ultracontractive estimations under general Sobolev inequality which imply in the particular case of a probability measure, transportation inequalities.
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 5 (1 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 . . . . . . . . . . . . . . . . .
Fast diffusion flow on manifolds of nonpositive curvature
 J. Evol. Equ
"... We consider the fast diffusion equation (FDE) ut = ∆u m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L p –L q smoothing effects of the type ‖u(t)‖q ≤ Ct −α ‖u0 ‖ γ p, ..."
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Cited by 4 (4 self)
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We consider the fast diffusion equation (FDE) ut = ∆u m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L p –L q smoothing effects of the type ‖u(t)‖q ≤ Ct −α ‖u0 ‖ γ p, the case q = ∞ being included. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−∆)> 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.
Jump processes and nonlinear fractional heat equations on fractals
 Math. Nachr
"... Abstract. Jump processes on metricmeasure spaces are investigated by using heat kernels. It is shown that the heat kernel corresponding to a σstable type process on a metricmeasure space decays at a polynomial rate rather than at an exponential rate as a Brownian motion. The domain of the Dirichl ..."
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Cited by 3 (3 self)
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Abstract. Jump processes on metricmeasure spaces are investigated by using heat kernels. It is shown that the heat kernel corresponding to a σstable type process on a metricmeasure space decays at a polynomial rate rather than at an exponential rate as a Brownian motion. The domain of the Dirichlet form associated with the jump process is a SobolevSlobodeckij space, and the embedding theorems for this space are derived by using the heat kernel technique. As an application, we investigate nonlinear fractional heat equations of the form ∂u ∂t (t, x) = −(−∆)σu(t, x) + u(t, x) p with nonnegative initial values on a metricmeasure space F, and show the nonexistence of nonnegative global solution if 1 < p ≤ 1 + σβ, where α is the Hausdorff dimension of α F whilst β is the walk dimension of F.