Estimates on the Green's function of second-order elliptic operators in RN

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equation (L + lambda)u = -Delta u + a(x) . del u + lambda u = f, x is an element of R-N, for lambda > 0. Initially, in a Cartesian frame, it is assumed that parallel to a(i)parallel to(L infinity(RN)) less than or equal to M-i, i = 1, ..., N. Pointwise estimates for the heat kernel of u(t) + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint parallel to\a\(t2)parallel to(L infinity(RN)) less than or equal to M is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on parallel to(L + lambda I)(-1)parallel to(L1(RN)) are derived as a consequence.