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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 ..."
Abstract

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
Mean Value Inequalities
"... 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 ..."
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Cited by 13 (4 self)
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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
Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry
"... 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 ..."
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Cited by 10 (4 self)
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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
Hölder Estimates And Regularity For Holomorphic And Harmonic Functions
, 2000
"... 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 # ..."
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Cited by 2 (0 self)
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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 [MS] 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 zplane and 0 on the wplane 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 [LT], one can also interp...
LOCAL GRADIENT ESTIMATE FOR pHARMONIC FUNCTIONS ON RIEMANNIAN MANIFOLDS
"... Abstract. For positive pharmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension n, p and the radius of the ball on which the function is de…ned. Our approach is based on ..."
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Abstract. For positive pharmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension n, p and the radius of the ball on which the function is de…ned. Our approach is based on a careful application of the Moser iteration technique and is di¤erent from ChengYau’s method [2] employed by Kostchwar and Ni [5], in which a gradient estimate for positive pharmonic functions is derived under the assumption that the sectional curvature is bounded from below. 1.