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Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
Abstract

Cited by 32 (2 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
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 26 (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
A NOTE ON POINCARI, SOBOLEV, AND HARNACK INEQUALITIES
"... differential operator with real cg(R) coefficients on M and such that L1 0 (i.e., L has no zeroorder term). Assume that there exists a positive cgoo measure # on M such that ..."
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differential operator with real cg(R) coefficients on M and such that L1 0 (i.e., L has no zeroorder term). Assume that there exists a positive cgoo measure # on M such that