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The Weyl calculus with locally temperate metric and weights. (English) Zbl 0621.47045
We extend the Weyl calculus of L. Hörmander [Commun. Pure Appl. Math. 32, 359-443 (1979; Zbl 0388.47032)] to symbols in \(T^*{\mathbb{R}}^ n\), which are temperate in the \(\xi\) variables. Thus they may grow exponentially in the x variables. In order to do that, we introduce a metric in the x variables, to define neighborhoods over which the symbols are temperate. We use cut-off functions \(\chi\) supported in the corresponding neighborhood of the diagonal, to define the operators \[ a^ w_{\chi}(x,D)u(x)=(2\pi)^{-n}\iint e^{<x-y,\xi >}\chi (x,y)a(\frac{x+y}{2},\xi)u(y)dyd\xi,\quad u\in C_ 0^{\infty}({\mathbb{R}}^ n), \] where a(x,\(\xi)\) is locally temperate. This generalizes V. I. Feijgin’s definition [Trudy Mosk. Mat. O.- va. 36, 155-194 (1978; Zbl 0421.35083)], and is independent of the choice of \(\chi\) modulo terms of arbitrary low order, if \(\chi\equiv 1\) in a neighborhood of the diagonal. We develop a calculus for these operators and show that they are continuous in \(C_ 0^{\infty}\) and \({\mathcal D}'\). We prove \(L^ 2\)-continuity when the symbol is bounded, and compactness when the symbol vanishes at \(\infty\). The Hilbert-Schmidt norm of the operator is bounded by the \(L^ 2\)-norm of the symbol, and we prove an estimate on the trace class norm.
Finally, we give an application of the calculus by improving and generalizing V. I. Feijgin’s estimate [Math. Sb., n. Ser. 99(141), 594-614 (1976; Zbl 0336.35079)] of the error term in the Weyl formula of the number N(\(\lambda)\) of eigenvalues \(\leq \lambda\) of certain self- adjoint \(p^ w_{\chi}\) in \({\mathbb{R}}^ n\), \[ N(\lambda)\sim (2\pi)^{-n}\iint_{p(x,\xi)\leq \lambda}dxd\xi, \] in the same way L. Hörmander [Ark. Mat. 17, 297-313 (1979; Zbl 0436.35064)] improved the estimate of V. N. Tulovskij and M. A. Shubin [Mat. Sb. n. Ser. 92(134), 571-588 (1973; Zbl 0286.35059)]. One example is the Laplacean with an exponentially growing real potential. For some temperated symbol classes sharper estimates for the error term are known, see V. I. Feijgin [Funkts. Anal. Prilozh. 16, No.3, 88-89 (1982; Zbl 0509.35077)] and references there.
Non-temperature metrics and weights also appear in higher order micro- localizations. For a more refined calculus for these symbol classes, see J. M. Bony and N. Lerner [Sém. éq. aux dér. part. 1986- 1987, No.2 and 3, École Polytechnique].
Reviewer: Reviewer (Berlin)

MSC:
47Gxx Integral, integro-differential, and pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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[9] Tulovskii, V. N. andŜubin, M. A., On the asymptotic distribution of eigenvalues of pseudodifferential operators inR n ,Mat. Sbornik,92 (134) (1973), 571–588 (Russian). English translation inMath. USSR Sbornik,21 (1973), 565–583.
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