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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
Stability of parabolic Harnack inequalities
 Transactions of the American Mathematical Society
, 2004
"... Abstract. Let (G;E) be a graph with weights faxyg for which a parabolic Harnack inequality holds with spacetime scaling exponent 2. Suppose fa0xyg is another set of weights that are comparable to faxyg. We prove that this parabolic Harnack inequality also holds for (G;E) with the weights fa0xyg. ..."
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Cited by 19 (3 self)
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Abstract. Let (G;E) be a graph with weights faxyg for which a parabolic Harnack inequality holds with spacetime scaling exponent 2. Suppose fa0xyg is another set of weights that are comparable to faxyg. We prove that this parabolic Harnack inequality also holds for (G;E) with the weights fa0xyg. We also give stable necessary and sucient conditions for this parabolic Harnack inequality to hold.
Some remarks on the elliptic Harnack inequality
, 2003
"... In this note we give three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of points in balls, and the second that the EHI is equivalent to an annulus type Harnack inequality f ..."
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Cited by 11 (0 self)
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In this note we give three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of points in balls, and the second that the EHI is equivalent to an annulus type Harnack inequality for Green’s functions. The third result uses the lamplighter group to give a counterexample concerning the relation of coupling with the EHI.
Heat kernels and sets with fractal structure, in Heat kernels and analysis on manifolds, graphs, and metric spaces
 Contemporary Math
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Divergence Form Operators on FractalLike Domains
, 2000
"... We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are preSierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for nonnegative Lharmonic functions on these domai ..."
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Cited by 7 (2 self)
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We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are preSierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for nonnegative Lharmonic functions on these domains and establish upper and lower bounds for the corresponding heat equation.
LOCAL SUBGAUSSIAN ESTIMATES ON GRAPHS: THE STRONGLY RECURRENT CASE
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