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37
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 38 (4 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,
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 32 (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.
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 30 (6 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.
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 27 (3 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 26 (2 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 26 (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.
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 whi ..."
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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.
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 asymptotic dimension for metric spaces, and the 0th NovikovShubin invariant
"... A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a cl ..."
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Cited by 6 (5 self)
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A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0th NovikovShubin number α0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos. 0. Introduction.