Abstract
Two algebras of global pseudo-differential operators over ℝn are investigated, with corresponding classes of symbols A0=CB∞ (all (x, ξ)-derivatives bounded over ℝ2n), and A1 (all finite applications of ∂xj, ∂ξj, and εpq=ξp∂ξq−p∂xp on the symbol are in A0). The class A1 consists of classical symbols, i.e., ∂ xα ∂ ξβ a= 0((1+|ξ|)−|α|) for x ∈ Kc ℝ;n, K, compact. It is shown that a bounded operator A of 210C=L2(Rn) is a pseudo-differential operator with symbol a∈Aj if and only if the map A→G−1AG, G∈ gj is infinitely differentiable, from a certain Lie-group gj c GL(210C) to ℒ(210C) with operator norm. g0 is the Weyl (or Heisenberg) group. Extensions to operators of arbitrary order are discussed. Applications to follow in a subsequent paper.
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References
BEALS, R.: A General Calculus of Pseudo-Differential Operators. Duke Math. Journal42, 1–42 (1975)
BEALS, R.: Characterization of Pseudo-Differential Operators and Applications. Duke Math. Journal44, 45–57 (1977)
BEALS, R.: On the Boundedness of Pseudo-Differential Operators. Comm. in Partial Diff. Equ. 210, 1063–1070 (1977)
CALDERON, A.; VAILLANCOURT, R.: On the Boundedness of Pseudo-Differential Operators. J. Math. Soc. Japan23, 374–378 (1971)
CALDERON, A.; VAILLANCOURT R.: A Class of Bounded Pseudo-Differential Operators. Pro. Nat. Acad. Sci. USA69, 1185–1187 (1972)
CORDES, H. O.: Lecture Notes on Banach Algebras and Partial Differential Operators. Lund 1970/71
CORDES, H. O.: On Compactness of Commutators of Multiplications and Convolutions, and Boundedness of Pseudo-Differential Operators. Journal of Functional Anal.18, 115–131 (1975)
CORDES, H. O.: Lecture Notes on Partial Differential Equations. Berkeley 1976/77
CORDES, H. O.; WILLIAMS, D.: An Algebra of Pseudo-Differential Operators with Non-Smooth Symbol. To appear in Pacific Journal Math. (1978/79)
HADAMARD, J.: Lectures on Cauchy's Problem in Linear Partial Differential Equations. New York: Dover Publ. inc. 1952
HELTON, J.; HOWE, R.: Traces of Commutators of Integral Operators. Acta. Math.136, 272–305 (1976)
HÖRMANDER, L.: Pseudo-Differential Operators and Hypoelliptic Equations. In: Singular Integrals, Proc. Symposia Pure Math. Vol. X (Chicago 1966), 138–183. Providence AMS 1967
HOWE, R.: Quantum Mechanics and Partial Differential Equations. To appear.
KATO, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer 1966
LOUPIAS, G.; MIRACLE-sOLE, S.: C*-algebres des Systemes Canoniques I. Commun. Math. Phys.2, 31–48-(1966)
LOUPIAS, G.; MIRACLE-SOLE S.: C*-algebres des Systemes Canoniques II. Ann. Inst. Henri. Poincare6, 39–58 (1967)
MAGNUS, W.; OBERHETTINGER, F.; SONI, R.: Formulas and Theorems for Special Functions of Mathematical Physics. Berlin-Heidelberg-New York: Springer 1966
RICKART, C. E.: General Theory of Banach Algebras. Princeton-Toronto-London-New York: van Nostrand 1960
SEGAL, I.: Transforms for Operators and Symplectic Automorphisms over a Locally Compact Abelian Group. Math. Scand.13, 31–43 (1963)
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Dedicated to Hans Lewy and Charles B. Morrey, Jr.
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Cordes, H.O. On pseudo-differential operators and smoothness of special Lie-group representations. Manuscripta Math 28, 51–69 (1979). https://doi.org/10.1007/BF01647964
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DOI: https://doi.org/10.1007/BF01647964