aequationes mathematicae

, Volume 40, Issue 1, pp 201–234 | Cite as

On realizations related to Weyl operators

  • J. Tervo
Research Papers
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Summary

The paper deals with the minimal and the maximal realizations (L w )~ and (L w )′•:L2→L2 of linear operators of Weyl type
$$(L{}^w(x,D)\varphi )(x) = (2\pi )^{ - n} \int_{\mathbb{R}^n } {\left( {\int_{\mathbb{R}^n } {L((x + y)/2,\xi )\varphi (y)e^{i< x - y,\xi > } dy} } \right)d\xi }$$
(1)
Applying the Weyl symbolic calculus, one establishes sufficient criteria for the equality (L w )~ = (L w )′#, that is, for the essential maximality ofL w (x, D) inL2. Moreover criteria are obtained for the bijectivity of (L w )~+ CI for large enoughC. When the underlying Riemannian metricg satisfies the conditiong (x,ξ) (y, η) = g (x,ξ) (y, − η), then the operators (1) are related to the pseudo-differential operators
$$(L(x,D)\varphi )(x) = (2\pi )^{ - n} \int_{\mathbb{R}^n } {L(x,\xi )(F\varphi )(\xi )e^{i< \xi ,x > } d\xi }$$
(2)
throughL(x, D) = A w (x, D), whereA(x, ξ) = e −i〈D x ,Dξ〉/2L(x, ξ). From this relation some properties ofL w (x, D) are carried over toL(x, D). For example, criteria forL~ =L′# are verified.

AMS (1980) subject classification

Primary 35S05 Secondary 35A35, 35A05 
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Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • J. Tervo
    • 1
  1. 1.University of KuopioKuopioFinland

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