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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.
Graphs between the elliptic and parabolic Harnack inequalities.
"... We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L ..."
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Cited by 13 (0 self)
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We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L 2 Poincaré inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass, see [2]. It satisfies the volume regularity and not the Poincaré inequality. We construct another example that does not satisfy the volume regularity.
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.
HARNACK INEQUALITY FOR A CLASS OF DEGENERATE ELLIPTIC OPERATORS
, 2003
"... Abstract. We prove a Harnack inequality for a class of twoweight degenerate elliptic operators. The metric distance is induced by continuous Grushintype vector fields. It is not know whether there exist cutoffs fitting the metric balls. This obstacle is bypassed by means of a covering argument tha ..."
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Cited by 2 (0 self)
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Abstract. We prove a Harnack inequality for a class of twoweight degenerate elliptic operators. The metric distance is induced by continuous Grushintype vector fields. It is not know whether there exist cutoffs fitting the metric balls. This obstacle is bypassed by means of a covering argument that allows the use of rectangles in the Moser iteration. 1.
Abstract L 2; � �Q�Estimates for Parabolic Equations and Applications
"... In this paper we derive L 2; � �QT �estimates for the �rst order derivatives of solutions to the following parabolic equation ut � @ �aij�x; t�uxj + aiu�+biuxi +cu = fi + f0; ..."
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In this paper we derive L 2; � �QT �estimates for the �rst order derivatives of solutions to the following parabolic equation ut � @ �aij�x; t�uxj + aiu�+biuxi +cu = fi + f0;