Riesz means of eigenfunction expansions of elliptic differential operators on compact manifolds

Let {λ ²} and {? _λ } be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator Δ on a compact manifoldM of dimensionN. We prove that the Riesz means of order δ, defined by (R_Lambda ^delta f = sumlimits_{lambda< Lambda } {left( {1 - frac{{lambda ^2 }}{{Lambda ^2 }}} right)^delta hat f(} lambda ) varphi _lambda ) , are uniformly bounded from the Hardy spaceH ^p (M) into Weak-L ^p (M), if 0<p<1 and δ=N/p?(N+1)/2.