Transactions of the American Mathematical Society · 1978 · 105 citations · 9 references
Natural SciencesMathematicsPhysicsMathematical PhysicsRepresentation TheoryTheoretical PhysicsGroup TheoryHilbert Space FormalismLie GroupsModular FormsOperator TheoryQuantum Field TheoryAnnotation EncodingDilation TheoryGeometric Differential OperatorsInvariant Differential OperatorsIrreducible RepresentationsLie Algebra ModulesLie Algebra So(3)Lie Algebra Theory
A representation-theoretic characterization is given for hypoellipticity of homogeneous (with respect to dilations) left-invariant differential operators <italic>P</italic> on the Heisenberg group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; it is the precise analogue for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the statement for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that a homogeneous constant-coefficient differential operator is hypoelliptic if and only if it is elliptic. Under these representation-theoretic conditions a parametrix is constructed for <italic>P</italic> by means of the Plancherel formula. However, these conditions involve all the irreducible representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P equals normal upper Sigma Subscript i equals 1 Superscript n Baseline upper X Subscript i Superscript 2 m Baseline plus upper Y Subscript i Superscript 2 m"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mi>i</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>Y</mml:mi> <mml:mi>i</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript i Baseline comma upper Y Subscript i Baseline comma i equals 1 comma ellipsis comma n"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{X_i},{Y_i},i = 1, \ldots ,n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <italic>Z</italic> generate the Lie algebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via the commutation relations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper X Subscript i Baseline comma upper Y Subscript j Baseline right-bracket equals delta Subscript i j Baseline upper Z"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>
9
Hypoelliptic second order differential equations
Lars Hörmander · Acta Mathematica · 1967 · 2.4K citations · Full text
Gerald B. Folland, E. M. Stein · Communications on Pure and Applied Mathematics · 1974 · 972 citations
UNITARY REPRESENTATIONS OF NILPOTENT LIE GROUPS
A. A. Kirillov · Russian Mathematical Surveys · 1962 · 690 citations