We show that, if a certain Sobolev inequality holds, then a scale-invariant 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

this article, we will discuss various issues concerning when a complete Riemannian manifold possesses a global mean value inequality for positive subsolutions of either the Laplace equation or the heat equation. This study is motivated by the recent result of the first author [L1]. In that paper, he proved estimates on the dimensions of spaces of harmonic functions of at most polynomial growth of degree

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this paper we will assume that (M; g) is a manifold of bounded geometry, i.e. the following two conditions are satised: (a) r inj > 0 where r inj is the radius of injectivity of M ; (b) jr

this paper is to prove (Theorem 4) that if a complete Kahler manifold satisfies the weak mean value property then an estimate of the type (0.1) holds. Moreover, if at a point the manifold has at most polynomial H OLDER ESTIMATES AND REGULARITY 3 volume decay for small balls, then it will have C # regularity (Corollary 5). On the other hand, if the manifold has at most polynomial volume growth for large balls, then it has the #- Liouville property (Corollary 8) for holomorphic functions. In fact, we will obtain a uniform C # estimate for all holomorphic functions (Corollary 5) if we assume a volume comparison condition. A typical situation to apply Theorem 4 is when the Kahler manifold is a singular algebraic or minimal variety. In this case, the subvariety inherits a Sobolev inequality from the ambient manifold [M-S] hence possesses the weak mean value property. However, in order to deal with this situation, the balls in Theorem 4 are usually taken to be extrinsic balls rather than geodesic balls. All the argument in Theorem 4 will carry through with the one exception that extrinsic balls are not necessarily connected. In order to overcome this, the assumption of local irreducibility (Definition 6) is imposed at the point in question. In fact, the local irreducibility assumption is necessary as can be seen by taking M as the union of the z and w planes in C 2 . The singular point p = (0, 0) disconnects the two copies of C hence violates the local irreducibility assumption. Moreover, the function given by 1 on the z-plane and 0 on the w-plane will be bounded holomorphic function on M that is discontinuous. Also note that since any complete manifold with Ricci curvature bounded from below possesses the weak mean value property locally [L-T], one can also interp...