Estimates for Riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds

Let {λ2} and {θ{symbol}λ} be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator on a compact N-dimensional manifold M. The Riesz means of order δ of an integrable function on M are defined by RΛδf{hook}(x) = ∝M∑λ < Λ ( 1 - λ2 Λ2)δθ{symbol}λ(x)θ{symbol}λ(y)f{hook}(y)dμ(y). In this paper we study the kernels and the operator norms of he operators {RΛδ} on Lp(M), 1 ≤ p ≤ 2(N + 1) (N + 3). We also prove that if 1 ≤ p < 2(N+ 1) (N + 3), and δ = N p - (N + 1) 2, then the operators {RΛδ} are of weak type (p, p) uniformly with respect to Λ. © 1991.