SHARP POINTWISE ESTIMATES ON HEAT KERNELS by ADAM SIKORA (WROC LAW) 1. Introduction. Let M be a connected and complete Riemannian manifold. By ρ we denote the Riemannian distance on M and by −L the closure of the Laplace-Beltrami operator on C ∞ c (M ) in L 2 (dx), where dx is the riemanian measure on M . For every bounded Borel function F : [0.∞) → IC , we define an operator F (L): L 2 (dx) → L 2 (dx) by the formula F (L)=  ∞ 0 F (λ) dE(λ), where E(λ) is the spectral decomposition of the operator L. We denote by K F (L) the kernel of the operator F (L) i.e. the distribution [F (L)(ψ)](x)=  M ψ(y)K F (L) (x,y) dy, where ψ ∈ C ∞ c (M ) and x,y ∈ M . Next for H t (λ) = exp(−tλ 2 ), we put p t (x,y)= K Ht ( √ L) (x,y). (1) The function p t (x,y) is called the heat kernel. Since L is symmetric with respect to dx, for any bounded Borel function F we have K F (L) (x,y)= K F (L) (y,x) (2) and p 2t (x,x)= ‖p t (x, ·)‖ 2 L 2 (dx) . On the present paper we consider the manifolds which satisfy the following assamption, which seems to appear first in Davies and Pang [3]. sup x∈M p t (x,x) ≤  Ct −d/2 if t ≤ 1 Ct −D/2 if t> 1. (3) Under this assumption Davies and Pang proved (cf. also [1]),that if d ≤ D then p t (x,y) ≤ C ′ min  t −d/2 (1 + ρ(x,y)/ √ t) d , (4) t −D/2 (1 + ρ(x,y)/ √ t) D  exp  −ρ(x,y) 2 4t  1