JOURNAL OF DIFFERENTIAL EQUATIONS 55, l-29 (1984) Kernels of Elliptic Operators: Bounds and Summability DAVID GURARIE * Department of Mathematics, Oregon State University, Cowallis, Oregon 97331 Received May 18, 1982; revised December 6, 1982 and April 18, 1983 In this paper we shall study the kernel of the resolvent R = (c-A)-’ and of some other related “functions of A” (e.g., e-“) for elliptic operators A on F?“, or more generally, for perturbations of elliptic operators. It turns out that the resolvent (consequently all other related functions) are given by an integral kernel, which is bounded by a convolution with a radial decreasing L ‘-function. This result has numerous applications: bounds for L”-spectrum of A, closedness, semigroup generation, essential selfadjointness, summability etc. In [GKl] we studied this problem for perturbation of constant- coefficient elliptic operators and established the following bound on the kernel R,(x, y) of ([--A)-‘, IR,(w)I ~c(W”‘W’” IX-VI); P = ICI7 (1) where H is the radial L ‘-function H(z) = i IZl-s; IZIG 1 lzl-‘; IzI> 1 (s < n < t). (4 Though the class of perturbations considered in [GKl ] was wide enough to include such examples, as Schrodinger operators with “Coulomb” and certain “magnetic” potentials (see [RS, Ch. lo]); -A + B(x) . V + V(x) V=q$+ Zj lXjtYXjf (Xi ER3)T the method of [GKl ] was limited to operators with only constant-coefficient leading part. The purpose of the present paper is to extend the results of [GK 1 ] to operators with variable coefficients, in particular, operators on manifolds. By doing so we also improve the type of bound (2) and show that for differential * Present address: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106. 1 0022-0396184 $3.00 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.