We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

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 paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable Laplacian. For x 2 # let us denote ae(x)=jxj =( P i x i ) . It is obvious that wehave the following two relations ) = 2n# (1.1) jraej = 1# x 6=0: (1.2) By integrating (1.1) over the ball B(r)ofradiusr centered at the origin, weobtain 2nVol(B(r)) = ) dVol(x)= @B(r) 2ae @ dA=2rA(@B(r)) where wehave used the fact that on the boundary @ = jraej = 1. Therefore, we have the following identity for the volume function V (r):=Vol(B(r)) V (r)= r (r): (1.3) Of course, the relation (1.3) of the volume and the boundary area of the Euclidean ball is well known from the elementary geometry.However, (1.1)-(1.2) can also be used in a rather sophisticated waytoprove the following isoperimetric inequality between the volume and the boundary area of any bounded (assume for simplicity that the boundary is smooth) A(@ cVol : (1.4) The constant c obtained in this way, is not the sharp one. As is well-known, the exact constant c in (1.4) is one for which both sides of (1.4) coincide for\Omega being a ball

We give upper estimates on the long time behaviour of the heat kernel on a non-compact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.

2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow

In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...

Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n and unbounded for p ≥ n if there is more than one end. It follows from known results that in such a case the Riesz transform on M is bounded for 1 < p ≤ 2 and unbounded for p> n; the result is new for 2 < p ≤ n. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in L p for some p> 2 for a more general class of manifolds. Assume that M is a n-dimensional complete manifold satisfying the Nash inequality and with an O(r n) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on L p for some p> 2 implies a Hodge-de Rham interpretation of the L p cohomology in degree 1, and that the map from L 2 to L p cohomology in this degree is injective. 1.

Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The p-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The non-smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 p-parabolicity and p-hyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and p-parabolicity . . . . . . . . . . . . . . . . .

Abstract. Let M be a compact n-dimensional manifold, n ≥ 2, with metric g(t) evolving by the Ricci flow ∂gij/∂t = −2Rij in (0, T) for some T ∈ R + ∪ {∞} with g(0) = g0. Let λ0(g0) be the first eigenvalue of the operator −∆g0 recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any n ≥ 2 when either T < ∞ or λ0(g0)> 0. As a consequence we extend Perelman’s local κ-noncollapsing result along the Ricci flow for any n ≥ 2 in terms of upper bound for the scalar curvature when either T < ∞ or λ0(g0)> 0. R(g0) 4 with respect to g0. We extend a Recently there is a lot of studies on Ricci flow on manifolds because it is an important tool in understanding the geometry of manifolds [H1–3], [Hs1–3], [KL], [MT], [P1], [P2]. On the other hand given any compact n-dimensional manifold M, n ≥ 2, with a fixed metric g it is known that Sobolev inequalities hold [He]. More specifically for any q ∈ [1, n) and p satisfying 1 p

We prove that a two-sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. Contents