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**On the convergence of Riesz means on compact manifolds.**
*(English)*
Zbl 0653.35068

Let M be a compact connected \(C^{\infty}\) manifold of dimension \(n\geq 2\) and P an elliptic differential operator on M with \(C^{\infty}\) coefficients and self-adjoint with respect to some positive \(C^{\infty}\) density dx, i.e. in the Lebesgue space \(L^ 2(M)\) associated with dx. Then \(L^ 2(M)\) has the direct sum decomposition \(L^ 2(M)=\sum^{\infty}_{j=1}E_ j\), where \(E_ j\) is a one- dimensional eigenspace of P corresponding to an eigenvalue \(\lambda_ j\). Let the eigenvalues of P be arranged in non-decreasing order and suppose, without loss of generality, that they are non-negative. If \(e_ j: L^ 2(M)\to L^ 2(M)\) denotes the projection operator onto \(E_ j\), the Riesz means of index \(\delta\geq 0\) of \(f\in D'(M)\) is defined as \(S_ L^{\delta}(f)=\sum^{\infty}_{j=1}(1-\lambda_ j/L)_+^{\delta}e_ j(f).\) For \(f\in L^ 2\), \(S_ L^{\delta}(f)\to f\) as \(L\to \infty\) for any \(\delta\geq 0\) but if \(1\leq p<2\) and \(f\in L^ p\) a necessary condition for \(S_ L^{\delta}(f)\to f\) in \(L^ p\) is that \(\delta >\delta (p):=\max (0,n| 1/p-| -).\) This criterion is also known to be sufficient in some cases, e.g. when M is the torus \(T^ n\) and P the standard Laplacian on \(T^ n.\)

There are two main results in this paper. Firstly it is proved that if \(f\in L^ 1(M)\) then \(S_ L^{\delta}(f)\to f\) in \(L^ 1\) if \(\delta >\delta (1)\); secondly when P is of order two and \(| 1/p-| \leq 1/n+1\) then for \(f\in L^ p\), \(S_ L^{\delta}(f)\to f\) in \(L^ p\) if \(\delta >\delta (p)\).

There are two main results in this paper. Firstly it is proved that if \(f\in L^ 1(M)\) then \(S_ L^{\delta}(f)\to f\) in \(L^ 1\) if \(\delta >\delta (1)\); secondly when P is of order two and \(| 1/p-| \leq 1/n+1\) then for \(f\in L^ p\), \(S_ L^{\delta}(f)\to f\) in \(L^ p\) if \(\delta >\delta (p)\).

Reviewer: W.D.Evans

### MSC:

35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |

58J32 | Boundary value problems on manifolds |