Riesz means of eigenfunction expansions of elliptic differential operators on compact manifolds. (English) Zbl 0723.35054
Let \(\lambda^ 2\), \(\phi_{\lambda}\) be the system of eigenvalues and orthonormalized eigenfunctions of the Laplace operator on a manifold M. The author proves that the Riesz means of order \(\delta\), i.e. \(R_{\lambda,\delta}f=\sum_{\lambda <\Lambda}[1-(\lambda^ 2/\Lambda^ 2)]\hat f(\lambda)\phi_{\lambda}\), are bounded in \(H^ p(M)\) with value in \(L^ p\)-weak(M) for \(0<p<1\) and \(\delta =N/p- (N+1)/2\).
Reviewer: M.Biroli (Monza)
MSC:
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 58J05 | Elliptic equations on manifolds, general theory |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
Riesz meansReferences:
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