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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

Citations:

Zbl 0653.35068
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References:

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