Solvability of left invariant differential operators on certain solvable Lie groups

Ohring, P A

Title: Solvability of left invariant differential operators on certain solvable Lie groups

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Abstract

In this work, the authors uses non-commutative harmonic analysis in the study of differential operators on a certain class of solvable Lie groups. A left-invariant differential operator (a differential operator that commutes with left translations on the group) can be synthesized in terms of differential operators on lower-dimensional spaces. This synthesis is easily described for a certain class of simply connected solvable Lie groups, those arising as semi-direct products of simply connected abelian groups. We derive sufficient conditions for the global solvability of left invariant differential operators on such groups in terms of the lower-dimensional differential operators. These conditions are seen to be satisfied for certain classes of second-order differential operators, thus yielding global solvability. Specifically elliptic, sub-elliptic, transversally elliptic and parabolic operators are investigated.

Authors:: Ohring, P A

Publication Date:: Thu Jan 01 00:00:00 EST 1987

Research Org.:: Colorado Univ., Boulder (USA)

OSTI Identifier:: 6984741

Resource Type:: Thesis/Dissertation

Resource Relation:: Other Information: Thesis (Ph. D.)

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; LIE GROUPS; MATHEMATICAL OPERATORS; DIFFERENTIAL EQUATIONS; MATHEMATICAL SPACE; NUMERICAL SOLUTION; EQUATIONS; SPACE; SYMMETRY GROUPS; 657000* - Theoretical & Mathematical Physics

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                    Ohring, P A. Solvability of left invariant differential operators on certain solvable Lie groups.  United States: N. p., 1987. 
        Web.

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                    Ohring, P A. Solvability of left invariant differential operators on certain solvable Lie groups.  United States.

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                    Ohring, P A. 1987.  
        "Solvability of left invariant differential operators on certain solvable Lie groups".  United States.

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@article{osti_6984741,

  title        = {Solvability of left invariant differential operators on certain solvable Lie groups},

  author       = {Ohring, P A},

  abstractNote = {In this work, the authors uses non-commutative harmonic analysis in the study of differential operators on a certain class of solvable Lie groups. A left-invariant differential operator (a differential operator that commutes with left translations on the group) can be synthesized in terms of differential operators on lower-dimensional spaces. This synthesis is easily described for a certain class of simply connected solvable Lie groups, those arising as semi-direct products of simply connected abelian groups. We derive sufficient conditions for the global solvability of left invariant differential operators on such groups in terms of the lower-dimensional differential operators. These conditions are seen to be satisfied for certain classes of second-order differential operators, thus yielding global solvability. Specifically elliptic, sub-elliptic, transversally elliptic and parabolic operators are investigated.},

  doi          = {},

  url          = {https://www.osti.gov/biblio/6984741},
  journal      = {},
number       = ,

  volume       = ,

  place        = {United States},

  year         = {Thu Jan 01 00:00:00 EST 1987},

  month        = {Thu Jan 01 00:00:00 EST 1987}

}

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