## The weak type $$L^ 1$$ convergence of eigenfunction expansions for pseudodifferential operators.(English)Zbl 0678.35096

Let $${\mathcal M}$$ be a smooth connected compact manifold without boundary of dimension $$n\geq 2$$ and let $${\mathcal P}({\mathcal D})$$ be a first-order elliptic pseudodifferential operator on $${\mathcal M}$$ which is selfadjoint with respect to a smooth positive density. Then $${\mathcal L}^ 2({\mathcal M})$$ admits a complete orthogonal direct sum decomposition with respect to the eigenspaces of $${\mathcal P}({\mathcal D})$$ and for $$f\in {\mathcal L}^ 2({\mathcal M})$$ it holds that $$f=\sum^{\infty}_{j=0}e_ j(f)$$ with convergence in the $${\mathcal L}^ 2$$-topology, where $$e_ j$$ denotes the projection on the j-th eigenspace.
This is no longer true in $${\mathcal L}^ p({\mathcal M})$$, $$p\neq 2.$$
However, for certain “averages” of the partial sums, for example for the Riesz means of index $$\delta >0$$ defined by ${\mathcal S}_ t^{\delta}f:=\sum_{\lambda_ j\leq t}(1-\lambda_ j/t)^{\delta}e_ j(f)\quad (\lambda_ j\quad are\quad the\quad eigenvalues\quad of\quad {\mathcal P}({\mathcal D}))$ this remains true. This was shown by the second author [Ann. Math., II. Ser. 126, 439-447 (1987; Zbl 0653.35068)].
The main result of the paper under review is the following. Let $${\mathcal L}^{1,\infty}({\mathcal M})$$ denote weak $${\mathcal L}^ 1({\mathcal M})$$ and let $$\delta =(n-1)/2$$. Then there exists a constant $${\mathcal C}$$ independent of t, such that $$\| S_ t^{\delta}f\|_{{\mathcal L}^{1,\infty}({\mathcal M})}\leq {\mathcal C}\| f\|_{{\mathcal L}^ 1({\mathcal M})},$$ e.g. a uniform weak $${\mathcal L}^ 1$$-type estimate holds.
Reviewer: J.Marschall

### MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators 47Gxx Integral, integro-differential, and pseudodifferential operators 58J40 Pseudodifferential and Fourier integral operators on manifolds

Zbl 0653.35068
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