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

Abstract

Let { λ 2} and { ϑ 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 R Λ δƒ(x) = ∝ M∑ λ < Λ ( 1 − λ 2 Λ 2) δ ϑ λ(x) ϑ λ(y) ƒ(y)dμ(y) . In this paper we study the kernels and the operator norms of he operators {R Λ δ} on L p(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 { R Λ δ } are of weak type ( p, p) uniformly with respect to Λ.