Results 1  10
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63
Sobolev inequalities in disguise
 Indiana Univ. Math. J
, 1995
"... We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff argu ..."
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Cited by 55 (8 self)
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We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff arguments. This method has interesting consequences concerning Trudinger type inequalities. 1. Introduction. On R n, the classical Sobolev inequality [27] indicates that, for every smooth enough function f with compact support,
Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
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Cited by 44 (5 self)
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We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 43 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 42 (5 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Higher Eigenvalues and Isoperimetric Inequalities on Riemannian manifolds and graphs
"... this paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable Laplacian. For x 2 # let us denote ae(x)=jxj =( P i x i ) . It is obvious that ..."
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Cited by 29 (3 self)
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this paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable Laplacian. For x 2 # let us denote ae(x)=jxj =( P i x i ) . It is obvious that wehave the following two relations ) = 2n# (1.1) jraej = 1# x 6=0: (1.2) By integrating (1.1) over the ball B(r)ofradiusr centered at the origin, weobtain 2nVol(B(r)) = ) dVol(x)= @B(r) 2ae @ dA=2rA(@B(r)) where wehave used the fact that on the boundary @ = jraej = 1. Therefore, we have the following identity for the volume function V (r):=Vol(B(r)) V (r)= r (r): (1.3) Of course, the relation (1.3) of the volume and the boundary area of the Euclidean ball is well known from the elementary geometry.However, (1.1)(1.2) can also be used in a rather sophisticated waytoprove the following isoperimetric inequality between the volume and the boundary area of any bounded (assume for simplicity that the boundary is smooth) A(@ cVol : (1.4) The constant c obtained in this way, is not the sharp one. As is wellknown, the exact constant c in (1.4) is one for which both sides of (1.4) coincide for\Omega being a ball
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 29 (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.
Pointwise estimates for transition probabilities of random walks in infinite graphs
 in: Trends in mathematics: Fractals in Graz 2001
, 2002
"... walks on infinite graphs ..."
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Anomalous heatkernel decay for random walk among bounded random conductances
, 2008
"... ABSTRACT. We consider the nearestneighbor simple random walk on Z d, d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy> 0 exceeds the threshold for bond percolation on Z d. For environments in ..."
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Cited by 13 (2 self)
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ABSTRACT. We consider the nearestneighbor simple random walk on Z d, d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy> 0 exceeds the threshold for bond percolation on Z d. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2nstep return probability P 2n ω (0,0). We prove that P 2n ω (0,0) is bounded by a random constant times n −d/2 in d = 2,3, while it is o(n −2) in d ≥ 5 and O(n −2 log n) in d = 4. By producing examples with anomalous heatkernel decay approaching 1/n 2 we prove that the o(n −2) bound in d ≥ 5 is the best possible. We also construct natural ndependent environments that exhibit the extra log n factor in d = 4. 1.
SaloffCoste L., Stability results for Harnack inequalities
"... We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Har ..."
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Cited by 10 (2 self)
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We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete noncompact manifolds having nonnegative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically nonnegative sectional curvature. Contents